THERMODYNAMICS 


OP 


KEVERSIBLE  CYCLES  IN  GASES 
AND  SATTJKATED  VAPORS, 


FULL  87NOP8I8   OF  A    TEN  WEEKS1    UNDERGRADUATE 
COURSE  OF  LECTURES  DELIVERED  BY 


M.    I.   PUPIN,   PH.D.      /K 


,  AND  ED 

?£rp9^^BEiia, 

Student  in  Electrical  Engineering,  Columbia  College 


YORK: 

EY    &    SONS, 
53  lUsT  TENTH  STREET. 
1894. 


5> 


p,  1894, 

BY 

MAX  OSTERBERG. 


PRINTER,  NFW  YORK. 


PREFACE. 


MY  DEAR  MR.  OSTERBERG: 

It  was  quite  a  pleasure  to  examine  your  carefully  worked 
out  notes  on  my  lectures.  I  agree  with  you  in  the  opinion  that 
the  publication  of  these  notes  will  be  of  much  service  to  your 
classmates  and  probably  to  others  who  may  be  interested  in 
the  elementary  features  of  thermodynamics. 

Very  sincerely  yours, 

M.  I.  PUPIN . 

COLUMBIA  COLLEGE,  NEW  YORK, 
December  20,  1893. 

Ill 


SD8        ^^j 

V1  T  I! 
i   X  II 


INTRODUCTION. 


IK  this  course  on  Theoretical  Thermodynamics  we  shall 
limit  our  discussion  to  those  features  of  the  science  which 
have  a  direct  bearing  upon  the  science  of  Caloric  En- 
gineering. The  course  forms,  therefore,  a  theoretical  in- 
troduction to  the  practical  course  on  Heat  Engines.  It 
seems  desirable,  however,  to  mould  our  discussion  in  such  a 
way  that  it  will  serve  at  the  same  time  the  very  important 
purpose  of  forming  an  introduction  to  the  study  of  some  of 
the  best  and  most  complete  works  on  the  subject.  The 
work  of  R.  Clausitis  (Die  meclianisclie  Wdrmetheorie)  is  and 
very  probably  will  always  remain  the  classical  treatise  on  this 
very  important  branch  of  exact  sciences.  We  shall,  therefore, 
adopt  the  mathematical  notation  and  f  ollow  as  closely  as  prac- 
ticable the  method  of  discussion  which  is  given  in  this  great 
work  of  Clausius,  who,  as  you  will  presently  see,  is  one  of  the 
principal  founders  of  the  beautiful  science  of  thermody- 
namics. 


THERMODYNAMICS 

OF 

REVERSIBLE  CYCLES  IN  GASES  AND 
SATURATED  VAPORS. 


NATURE    OF    HEAT    AND  THE  MATHEMATICAL  STATE- 
MENT OF  THE  FIRST  LAW  OF  THERMODYNAMICS. 

The  Science  of  Thermodynamics  investigates  the  formal 
or  quantitative  laws  which  underlie  all  physical  processes  by 
which  the  caloric  state  of  material  bodies  is  changed. 

By  physical  process  we  simply  mean  any  one  of  the  almost 
infinite  variety  of  ways  by  which  nature  transforms  one  form 
of  energy  into  another,  or  one  form  of  matter  into  another. 
Heat  is  developed  in  almost  every  physical  process,  owing 
principally  to  the  presence  of  passive  or  frictional  resistances, 
unavoidable  in  material  systems. 

The  first  question  to  be  considered  is :  What  is  heat  f  It 
certainly  must  be  one  of  the  two  fundamental  physical 
quantities,  that  is,  it  must  be  either  a  form  of  matter  or  a 


2  THERMODYNAMICS  OF 

form  of  energy.  In  the  first  case  it  will  be  subject  to  the 
principle  of  conservation  of  matter,  and  in  the  second  case  it 
will  be  subject  to  the  principle  of  conservation  of  energy. 

According  to  the  old  hypothesis  heat  is  a  substance  ;  but 
since  the  weight  of  a  body  is  independent  of  its  temperature, 
the  heat  substance,  the  so-called  Caloric,  had  to  be  supposed  to 
be  imponderable. 

The  experiments  of  Sir  Humphry  Davy  (1799*)  and  of 
Rumford  (1798  f)  showed,  however,  that,  1st,  the  latent  heat 
contained  in  a  given  body  could  be  increased,  and,  2d,  the 
temperature  of  a  body  could  be  changed  without  a  change  in 
the  quantity  of  heat  in  the  surrounding  bodies. 

The  first  was  shown  by  Davy's  experiment  of  melting  two 
pieces  of  ice  by  rubbing  them  one  against  the  other  ;  the  sec- 
ond was  shown  by  Rumford's  experiment,  in  which  heat  was 
developed  by  friction,  which  accompanies  the  boring  of 
metals.  These  experiments  gave  the  death-blow  to  the  old  or 
substance  theory  of  heat. 

They  showed  not  only  that  heat  cannot  be  a  substance,  but 
that  on  the  contrary  it  has  all  the  physical  characteristics  of 
the  other  fundamental  physical  quantity,  which  we  call  energy. 
As  such  it  must  be  subject  to  the  principle  of  conservation  of 
energy.  Joule  was  the  first  to  shoiv  that  heat  and  mechanical 
work  are  not  only  convertible  into  each  other,  but  that  the 

*  Essay  on  Heat,  Light,  and  the  Combinations  of  Light,  with  a  new 
Theory  of  Respiration.  Experiments  II.  and  III. 

f  Count  Rumford  *s  essay  on  "An  Inquiry  concerning  the  Source  of 
Heat  which  is  excited  by  Friction,"  read  before  the  Royal  Society, 
January  25th,  1798.  See  Memoir  of  Sir  Benjamin  Thompson,  Count 
Rumford,  with  notices  of  his  Daughter,  by  George  Edward  Ellis. 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS. 

ratio  of  conversion  is  constant  (1839-1844).  Later  on  he  ex- 
tended this  relation  to  other  forms  of  energy,  like  chemical 
energy,  etc.  By  these  experiments  Joule  not  only  proved 
that  heat  is  a  form  of  energy  convertible  into  mechanical  or 
any  other  form  of  energy  at  a  constant  ratio,  but  he  also  fur- 
nished the  first  exact  experimental  basis  for  the  general  prin- 
ciple of  conservation  of  energy,  first  enunciated  by  Julius 
Robert  Mayer  about  the  same  time  (1842).  It  was,  however, 
clearly  stated  in  its  most  general  form  and  mathematically 
elucidated  by  Helmholtz  in  1847. 

The  principle  of  Conservation  of  Energy  can  be  stated 
as  follows  :  Energy  cannot  be  created  nor  can  it  be  annihi- 
lated by  any  physical  processes  which  our  mind  can  con- 
ceive. The  energy  contained  in  an  isolated  material  system 
must,  therefore,  remain  constant,  although  all  sorts  of 
changes  may  be  taking  place  within  the  system.  These 
changes,  however,  will  consist  in  a  transformation  of  one 
form  of  energy  into  another  or  one  form  of  matter  into 
another  without  loss. 

When  the  energy  of  a  body  or  system  of  bodies  is  changed 
in  amount,  then  the  change  is  due  to  the  transference  of 
energy  from  external  bodies  to  the  system  considered,  or  vice 
versa.  In  the  first  case,  work  is  done  upon  the  system;  in  the 
second,  the  system  does  work  upon  external  bodies. 

If  we  heat  a  body,  one  part  of  the  heat  energy  put  into  it 
will  appear  as  heat,  another  part  as  work  done  against  ex- 
ternal forces,  like  surface  pressure,  and  another  part  as 
potential  energy  overcoming  the  internal  forces,  like  forces  of 
cohesion,  chemical  affinity,  etc.  The  common  unit  of  meas- 
ure of  all  these  forms  of  energy  is  the  unit  of  mechanical 


4  THERMODYNAMICS  OF 

work,  that  is,  the  kilogramme-meter,  The  unit  of  heat  is  one 
Tdgr.  calorie. 

DEF. — One  Tdgr.  calorie  is  the  heat  necessary  to  raise  the 
temperature  of  a  Iclgr.  of  a  standard  body  one  degree.  The 
standard  body  being-  pure  distilled  water  at  4°  C.,  and  at  nor- 
mal pressure  (760  mm.). 

The  mechanical  equivalent  of  heat  is  the  number  of  Tdgr. 
meters  which  must  be  spent  to  generate  that  much  heat. 

It  was  found  by  experiment  that  one  klg.  calorie  =  424 
klg.  m.  (very  nearly).  The  most  distinguished  experimental- 
ists in  this  line  being  Kobert  Mayer,  Joule,  Him,*  Kowlaud. 

To  express,  say,  890  klgr.  meters  in  calories,  divide  890  by 
one  J  ("  Joule  ")  or  424  (/  being  the  symbol  of  the  mechani- 
cal equivalent  of  heat). 

STATEMENT  OF  THE  FIRST  LAW  OF  THERMODYNAMICS  FOR 
A  PARTICULAR  CLASS  OF  PHYSICAL  PROCESSES. 

"We  will  now  limit  ourselves  to  that  class  of  bodies  and 
processes  in  which  the  external  work  done  is  mechanical  work, 
and  in  that  class  of  processes  we  shall  select  those  processes 
where  the  mechanical  work  is  done  in  overcoming  a  certain 
amount  of  pressure  on  the  surface  of  a  body  to  which  we 
apply  the  heat,  and  consider  that  the  pressure  is  uniform  and 
normal  at  all  points  of  the  surface. 

*Hirn,  Recherches  sur  1'equivalent  mecanique  de  la  chaleur,  p.  20; 
Also,  Theorie  mecanique  de  la  chaleur,  2d  edition,  part  1st,  p.  35. 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.         5 

Expression  for  External  Work  due  to  Infinitesimal  Expansion. 

Consider  an  infinitesimal  area  of  the  surface  du  square 
meters  ;  if  the  pressure  per  sq.  m.  is  p  kg.,  the  total  pressure 
is  pdu.  Now  suppose  the  body  to  expand,  overcoming  the 
surface  pressure :  the  work  done  by  the  act  of  displacing  du 
is  equal  to  the  normal  displacement  dn  meters  times  the 
pressure,  that  is  pdudn  kg.  meters. 

To  find  the  total  work  done  over  the  whole  surface,  we 
must  integrate  (remembering  that  p  is  uniform  all  over  the 
surface). 


/  pdudn  —pi  dudn  —  pdv. 


That  is  to  say,  for  infinitely  small  increments  of  volume,  dur- 
ing which  the  surface  pressure  may  be  considered  constant, 
the  external  work  is  equal  to  the  surface  pressure  multiplied 
by  the  increment  in  volume. 

Intrinsic  Energy  and  Co-ordinates  of  a  Body. 

Let  us  now  communicate  an  infinitely  small  quantity  of 
heat  dQ  to  a  body.  Experience  tells  us  that  the  following 
changes  in  the  body  may  be  looked  for:  change  in  tempera- 
ture, internal  aggregation,  chemical  constitution,  electrical 
and  magnetic  state,  volume,  etc.,  etc.  If  we  now  call  the 
energy  which  a  body  contains  in  consequence  of  its  tempera- 
ture, internal  aggregation,  chemical  constitution,  electrical  and 
magnetic  state,  etc.,  etc.,  the  intrinsic  energy  of  the  body,  and 
denote  it  symbolically  by  the  letter  U,  and  if  we  denote  by  W 


6  THERMODYNAMICS  OF 

the  external  work  which  a  body  does  against  the  surface 
pressure  in  expanding  from  a  given  volume  to  some  other 
volume,  then  it  is  evident  that  the  above  changes  in  tempera- 
ture, aggregation,  chemical  constitution,  electrical  and  mag- 
netic state,  volume,  etc.,  etc.,  imply  a  certain  small  change 
dU  in  the  intrinsic  energy  and  a  certain  small  external  work 
dW.  According  to  the  Principle  of  Conservation  of  Energy 
the  mechanical  equivalent  of  dQ  is  equal  to  the  mechanical 
equivalent  of  dU  plus  the  mechanical  equivalent  of  dW. 
Hence 


This  is  the  most  general  form  of  the  first  law  of  thermody- 
namics. It  is  a  formal  statement  of  the  Principle  of  Conser- 
vation of  Energy  for  processes  which  involve  a  transformation 
of  a  certain  amount  of  heat. 

Experimental  Physics  and  Experimental  Chemistry  teach 
us  how  to  measure  the  physical  quantities  by  means  of  which 
we  describe  the  various  states  of  physical  bodies,  these  quan- 
tities being  temperature,  forces  of  cohesion,  chemical  affinity, 
quantities  of  various  chemical  elements  contained  in  a  body, 
magnetic  and  electrical  intensity  in  the  various  parts  of  the 
body,  volume,  pressure,  etc.,  etc.  The  energy  contained  in  a 
body  depends  on  its  state  and  not  on  the  processes  by  means 
of  which  the  body  has  been  brought  into  that  state.  Hence 
the  physical  quantities  just  mentioned  which  describe  the 
state  of  the  body  will  also  determine  completely  the  value  of 
the  intrinsic  energy  U  which  the  body  has  in  that  state.  Tit  is 
is  what  we  mean  when  we  say  that  U  is  an  analytical  function 
of  the  above  physical  quantities. 


REVERSIBLE  CYCLES  IN  OASES  AND    VAPORS.         7 

The  external  work  which  a  body  does  against  surface  press- 
ure while  expanding  from  a  given  volume  to  some  other 
volume  may  or  may  not  be  independent  of  the  process  by 
means  of  which  this  expansion  is  effected.  To  be  general,  we 
shall  suppose  that  it  is  not  independent,  that  is  to  say,  we  shall 
suppose  that  W  is  such  a  function  of  the  physical  quantities 
which  define  the  state  of  the  body,  that  in  passing  from  one 
set  of  values  of  these  quantities  to  another  set  the  change  in 
the  value  of  JFwill  depend  on  the  physical  processes  by  means 
of  which  we  passed  from  the  first  to  the  second  set. 

This  will  be  illustrated  further  on  by  actual  examples.  The 
physical  quantities,  like  temperature,  magnetic  and  electrical 
intensity,  pressure,  volume,  chemical  affinity,  quantity  of  mat- 
ter, etc.,  which  define  the  state  of  a  body  are  called  the  co- 
ordinates of  a  body.  We  say  then  that  £7 and  JFare  functions 
of  the  co-ordinates  of  the  body. 

It  is  evident  that  the  more  complex  the  state  of  a  body  the 
more  co-ordinates  shall  we  need  to  describe  its  state.  For 
the  present  we  shall  limit  ourselves  to  those  physical  bodies 
and  physical  processes  in  which  temperature,  volume,  and 
pressure  suffice  to  describe  completely  the  slate  of  the  body, 
and  the  processes  by  means  of  which  that  state  is  varied. 

U  will  therefore  be  a  function  of  the  pressure  p,  the 
volume  v,  and  the  temperature  t ;  the  same  is  true  of  W. 

We  shall  measure  p  in  kilogrammes  per  square  meter,  v 
in  cubic  meters,  and  t  in  degrees  centigrade. 

It  must  be  observed,  however,  that  these  three  co-ordi- 
nates of  the  body  are  not  independent  of  each  other.  Thus 
the  surface  pressure  and  the  temperature  of  a  body  being 
given,  its  volume  is  also  determined.  Varying  thr  nTBgiiirp  ^ 

-  -'  -: :~  • 


'2 


8  THERMODYNAMICS  OF 

without  varying  the  temperature,  we  can  pass  from  any  volume 
to  any  other  volume  (within  certain  definite  limits). 

We  can  therefore  express  U  in  terms  of  two  of  these.  In 
what  follows  U  will  be  considered  a  function  of  t  and  v  unless 
the  contrary  is  stated. 

We  shall  therefore  have  in  the  case  of  these  bodies  and 
these  physical  processes 


Since  the  pressure  is  normal  and  uniform  and  constant  for 
infinitely  small  variations  of  volume  and  temperature, 

dW=pdv; 


Hence  the  first  law  of  thermodynamics  in  the  case  of  bodies 
and  processes,  which  we  are  considering,  says  :  If  we  com- 
municate an  infinitely  small  amount  of  heat  to  a  body 
under  pressure,  a  part  of  the  heat  is  used  up  in  increasing 
the  internal  energy  of  the  body,  part  of  which  is  potential 
and  part  heat  energy,  and  the  rest  appears  as  mechanical  work 
done  against  the  surface  pressure. 

If  a  finite  quantity  of  heat  Q  is  communicated  to  a  body, 
causing  the  same  to  pass  from  a  given  state,  in  which  its  tem- 
perature is  tl  ,  surface  pressure  pl  ,  and  total  volume  #,  ,  to  an- 
other state,  in  which  these  quantities  are  £2  ,  pa  ,  #2  ,  then 


Q=U,-Ut 


REVERSIBLE  CYCLES  IN  GASES  AtfD    VAPORS.         9 

That  is,  the  total  quantity  of  heat  is  equal  to  the  total  in- 
crement of  internal  energy  plus  the  total  external  work.  The 
increment  in  internal  energy  is  due  to  two  changes:  1st, 
change  of  temperature;  2d,  change  in  the  arrangement  of 
matter  in  the  body.  The  first  change  produces  a  Tariation 
in  the  thermometric  heat;  the  second,  in  the  latent  heat  of  the 
body. 

If  pressure  is  constant,  then  external  work  =p(v9  —  vt). 
But  since  this  is  not  always  the  case,  we  must,  in  order  to 
find  the  numerical  value  of  the  integral  fpdv,  consider  first 
the  law  of  variation  of  p  with  v. 

For  infinitely  small  variations  of  volume  the  pressure  can 
be  considered  constant. 

II. 

APPLICATION  OF  THE  FIRST  LAW  OF  THERMO- 
DYNAMICS TO  PERFECT  GASES. 

We  pass  now  to  the  application  of  the  First  Law  of  Ther- 
modynamics to  the  study  of  the  simplest  thermodynamic 
processes  that  can  be  performed  upon  the  simplest  class  of 
physical  bodies,  that  is,  we  pass  to  the  application  of  the  First 
Law  of  Thermodynamics  to  the  study  of  the  process  of  ex- 
pansion and  compression  of  perfect  gases. 

These  bodies  may  justly  be  considered  the  simplest  bodies, 
because  experimental  research  tells  us  that  their  co-ordinates 
describe  the  state  of  these  bodies  by  the  simplest  formal  rela- 
tions that  we  know  of,  namely, 

Relation  (1)  contains  what  is  known  as  Boyle's  law,  viz.  : 


That  is  to  say,  if  a  given  quantity  of  a  perfect  gas,  say  a  kilo- 


10  THERMODYNAMICS  OF 

gramme  of  dry  air  at  a  given  temperature,  occupies  a  volume 
vt  when  it  is  under  a  pressure  pt ,  then  the  same  quantity  of 
the  gas  at  the  same  temperature  when  it  is  under  pressure  p^ 
will  occupy  the  volume  va  as  given  in  above  equation. 

This  relation  is  true  lor  a  large  number  of  gases  within  a 
large  interval  of  pressure  and  temperature. 

Relation  (2)  contains  what  is  known  as  the  Mariotte-Gay- 
Lussac  law,  viz. : 

pv=pj>.(l  +  ot) (2) 

Here  p  is  pressure  in  kg.  per  square  meter  and  v  is  volume 
in  cubic  meters  of  a  unit  weight  (that  is  a  kilogramme)  of 
a  perfect  gas  when  it  is  under  the  pressure  p  and  when  its 
temperature  is  t  deg.  centigrade  above  the  standard  freezing- 
point.  pQ  and  v0  denote  the  corresponding  quantities  of 
the  same  weight  of  the  gas  at  the  freezing-point,  a  is 
called  the  temperature  coefficient  of  expansion.  It  is  found 
to  be  very  nearly  constant  for  all  temperatures  within  a  large 
interval,  and  to  be  =  -^^  (very  nearly)  for  all  perfect  gases. 

The  Mariotte-Gay-Lussac  law  can  also  be  written 

w  —  _£°V973   i   A  —  £2^0  T      RT 
P         273  l  '       273 

Here  T  is  called  the  absolute  temperature  of  the  gas  and  R 
is  a  constant  which  depends  on  the  nature  of  the  gas.  It  can 
.be  calculated  for  every  gas  from  experimental  data,  as  will  be 
shown  presently. 

Choosing  Tand  v  as  the  co-ordinates  of  the  gas,  we  have 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.       11 

ON  THE  RATES  OF  VARIATION  OF  THE  INTRINSIC  ENERGY 
OF  A  PERFECT  GAS. 

When  a  small  quantity  of  heat  dQ  is  communicated  to  a 
body,  then  a  part  of  it  will  appear  as  an  increment  in  the  in- 
trinsic energy  of  the  body,  and  the  other  part  will  appear  as 
external  work. 

The  first  part  consists  of  two  terms,  namely,  —-~  dT  and 

— dv.     Now— ™^jPis  the  increment  of  intrinsic  energy  due 
to  the  increase  of  the  body's  temperature  by  dT.     The  factor 

—7^  is  the  rate  at  which  internal  energy  varies  with  the  teln- 
et 

perature,  the  volume  being  constant.  If  that  rate  were  con- 
stant, then  the  factor  — =  would  mean  the  increment  of 

O-L 

internal  energy  due  to  a  rise  of  temperature  of  1°.  This 
part  -^TpdT  appears  as  sensible  heat  of  the  body,  i.e.,  heat 
which  can  be  measured  by  a  thermpmeter.  The  other  part 

^  TJ 

-^—dv  is  the  increment  of  intrinsic  energy  due  to  the  change 
of  the  body's  volume,  the  temperature  remaining  constant. 

The  factor  --—  is  the  rate  at  which  this  increment  takes 

dv 

place;    if   this    rate   remained  constant  for  finite  variations' 

of  volume,  then would  mean  the  increment  of  intrinsic 

'dv 

energy  for  a  unit  increment  in  volume,  the  temperature  re- 
constant^ 


12  THERMODYNAMICS  OF 

We  have  every  evidence  in  favor  of  the  hypothesis  that 
heat  is  a  mode  of  motion  of  the  molecules  of  a  body.  Its 
temperature  is  due  to  the  kinetic  energy  of  this  motion; 

hence  ^p^dT  denotes  the  increment  of  the  kinetic  part  of  the 

O-i 

intrinsic  energy.  The  other  part  of  the  increment  of  the  internal 

o  77" 
energy,  namely,  ——dv,  will  be  work  done  against  the  forces 

between  these  molecules.  These  forces  will  depend  on  the 
distances  of  the  molecules  from  each  other,  and  therefore  on 
the  volume  of  the  body.  The  work  done  against  these  forces, 
(when  the  temperature,  and  consequently  the  kinetic  energy  of 
the  body,  remains  constant),  will  therefore  appear  as  potential 
energy.  Whenever  there  are  no  internal  forces  between  the 

molecules,  the  term  -—dv  will  be  zero.    That  is,  the  intrin- 

dv 

sic  energy  of  the  body  will  be  independent  of  the  volume. 

If  such  a  body  expands  without  doing  work  against  external 
pressure,  its  temperature  will  remain  constant,  if  no  heat  is 
communicated  from  without.  For,  from  the  equation 


we  see  that  when  dQ  =  0,  there  being  no  heat  communicated 
from  without,  and  pdv  =  0,  there  being  no  external  work 
done,  then 


-dT==  0.    That  is,  dT  =  0,  since      -  cannot  =  0. 


REVERSIBLE  CYCLES  IN  OASES  AND   VAPORS.       13 

Vice  versa,  if  when  the  body  expands  without  doing  external 
work,  and  without  receiving  heat  from  without,  its  tempera- 
ture remains  constant,  then 


<fo=0.    Hence 

dv 


That  is  to  say,  the  intrinsic  energy  of  such  a  body  will  be  in- 
dependent of  its  volume. 

Gay-Lussac's  experiments  disclosd  such  a  behavior  on  the 
part  of  perfect  gases.  These  experiments  were  verified  later 
by  Joule  and  Eegnault.  We  conclude,  therefore,  that  for 
perfect  gases  the  first  law  of  thermodynamics  will  have  the 
following  form : 


A  perfect  gas  may  be  described  as  a  body  which,  when  expand- 
ing without  doing  external  work*  will  neither  cool  nor  heat. 

THE  VARIOUS  MATHEMATICAL  FORMS  OF  STATEMENT  OF 
THE  FIRST  LAW  FOR  PERFECT  GASES. 

7^  TT 

The  rate  — =  has  a  definite  name  in  thermodynamics.  Sup- 
pose that  the  quantity  of  gas  under  consideration  is  one  unit 
weight,  i.e.,  one  kilogramme;  --=•  means  then  the  quantity  of 

heat,  measured  in  mechanical  units,  which  we  would  have  to 
communicate  to  one  unit  weight  of  the  gas  at  constant  vo 


14  THERMODYNAMICS  OF 

in  order  to  raise  its  temperature  by  one  degree,  if  the  rate  at 
which  heat  is  supplied  during  a  rise  of  temperature  were 
constant.  This  quantity  of  heat  is  called  the  specific  heat  of 
the  gas  at  constant  volume,  and  is  denoted  by  Cv. 


since  p  =  --    and    —^  =  Cv)  we  can  write  down  the  first 
law  of  thermodynamics  for  perfect  gases  as  follows  : 

.   .....     (1) 


From  equation  (1)  we  can  eliminate  v  or  T  by  means  of 
the  relation  pv  =  RT,  and  obtain 


-—  dp     ....     (2) 

and 

n  n  _j_  7? 

<v.    ....    (3) 


Equations  (1),  (2),  (3)  represent  three  ways  of  stating  mathe- 
matically the  first  law  of  thermodynamics  in  the  case  of  a 
perfect  gas.  They  differ  from  each  other  only  in  the  selection 
of  the  co-ordinates. 

These  expressions  contain  two  physical  constants  of  the  gas, 
viz.,  Cv  =  specific  heat  at  constant  volume,  and  R.  Both 
these  constants  must,  of  course,  be  determined  experimentally. 
The  quantity  Cv  being  difficult  to  determine  experimentally, 
it  is  preferred  to  put  these  formulae  into  a  different  form, 
containing  Cp  —  specific  heat  at  constant  pressure. 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.       15 

To  bring  out  the  meaning  of  this  physical  constant,  and 
show  its  relation  to  CV9  suppose  that  we  have  a  unit  weight 
Of  air  enclosed  in  a  cylinder  by  means  of  a  piston,  which  can 
glfde  up  and  down  the  cylinder  without  friction.  Its  tem- 
perature, volume,  and  pressure  being  T,  v,  and  p,  we  must" 
havej?y  =  RT.  Clamp  the  piston,  and  apply  heat  to  the  gas 
until  its  temperature  rises  by  dT.  Let  dQl  be  the  mechani- 
cal measure  of  this  heat  communicated  to  the  gas  during  this 
operation;  then 


dQ,  =      ,dT=  Gv  dT. 

Now  loosen  the  clamp,  the  volume  will  increase,  since  the 
temperature  is  now  T  -\-  dT.  During  this  second  operation 
communicate  heat  to  the  expanding  gas  so  as  to  keep  its  tem- 
perature constant. 

The  volume  will  expand  from  v  to  v-\  —  -^dT,  overcoming 

pressure^.  Observe  that  —  ™  represents  the  rate  of  variation  of 

volume  due  to  variation  of  temperature,  the  pressure  being 
constant.  Hence  from  pv  =  BTvre  obtain 


Let  dQ^  be  the   heat   communicated   during  this  second 
operation-   then 


^  ((\-\-H\dT. 


16  THERMODYNAMICS   OF 

At  the  end  of  these  two  operations  we  have  a  change  in  tem- 

perature =  d  T,  a  change  in  volume  =  ^pdT,  and  external 

work  equal  to  p—~dT. 

ol 

Consider  now  another  process,  by  means  of  which  the  same 
change  in  temperature,  the  same  change  in  volume,  and  the 
same  external  work  is  produced.  Let  us  call  the  rate  at 
which  heat  must  be  applied  to  the  unit  weight  of  the  gas, 
when  it  expands  at  constant  pressure,  Cp.  Apply  heat  until 
the  same  change  in  volume  is  produced  as  before;  the  final 
temperature  will  be  T  -\-  dT,  as  before.  The  pressure  in  the 
second  case  being  the  same  as  in  the  first,  we  shall  have  the 
same  external  work  done  during  the  expansion,  and  the  in- 
crement in  internal  energy  will  also  be  the  same,  since  the 
change  of  temperature  is  the  same;  hence  the  quantity  of 
heat  communicated  will  also  be  the  same.  Call  it  dQ  j  then 

dQ  =  CpdT. 

But  since  dQ  =  dQl  -f-  dQ^  ,  we  must  also  have 
Cp  =  Cv  -f  R    or    Cv=  Cp-  R. 
Substituting  in  equations  (1),  (2),  (3),  we  obtain 


dQ=CpdT-     -dp;    ......     (2") 

P  * 


(3°) 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.       17 

THE  RATIO  OF  SPECIFIC  HEATS. 

C 
We  will  now  introduce  a  new  constant,  namely,  -~  =  k. 

Cv 

This  physical  constant  of  a  gas  was  first  introduced  by  Reg- 
nault,  and  is  of  vital  importance,  for  the  reason  that  k  can  be 
determined  experimentally  with  great  accuracy  by  determin- 
ing the  velocity  of  sound  through  the  gas. 

Regnault  found  k  for  air  =  1.41. 

Op  we  can  also  determine  by  experiment;  hence  we  can  cal- 
culate R. 

Vice  versa,  if  R  and  C^are  determined  experimentally,  then 
both  Cw  and  k  can  be  calculated. 

The  relation  Gp  —  R  4-  Cv ,  as  it  stands,  applies  to  values  of 
these  physical  constants  measured  in  mechanical  units.  Divid- 
ing by  J — the  mechanical  equivalent  of  heat — we  obtain 


or 


cp  =  7 


where  cp  and  cv  are  measured  in  kg.  calories. 

CALCULATION  OF  THE  MECHANICAL  EQUIVALENT  OF  HEAT. 

An  interesting  relation  is  obtained  from  the  last  equation. 

C  Rk 

Remembering,  namely,  that  -^  —  k,  we  obtain  J '  —  T — 

Uv  (K  —   \)(; 


18  THERMODYNAMICS  OF 

This  is  the  relation  first  pointed  out  by  Robert  Mayer,  and 
actually  employed  by  him  to  calculate  J. 

Before  we  can  apply  it  to  any  gas,  we  must  find  the  value 
of  R.     Let  us  find  the  same  for  air. 


273* 

Regnault  found  that  at  the  barometric  pressure  of  760  mm. 
the  value  for  v0  of  one  unit  weight  of  air  is  .7733  cubic 
meters.  But  since  our  units  are  kilogrammes  and  meters,  we 
have  to  change  the  above  value  of  p  accordingly. 

The  specific  weight  of  mercury  used  by  Regnault  is  13.596; 
we  get  therefore 

po  =  100°dw  X  7.6dwl  X  13.596  =  10333  kg.  per  square  meter. 
Hence,  since  VQ  =  .7733  cubic  meter, 

R  -  10333  X  .7733  _ 
273 

The  values  of  k  and  cp  for  air  were  determined  by  Reg 
nault.     They  are  k  =  1.41,  cp  =  .2375.     Hence  we  obtain 


T     29.27  X  1.41 
J= 


The  value  of  J  obtained  by  direct  experiments  is  very  nearly 
424.     So  much  for  the  physical  constants  of  a  perfect  gas, 


REVERSIBLE  CYCLES  IN  GASES  AND   VAPORS.      19 

A  table  of  cv  and  cp  for  various  gases  will  be  given  pres- 
ently. 

To  sum  up:  The  mathematical  statement  of  the  first  law 
of  thermodynamics  in  the  case  of  perfect  gases  expresses  a 
quantitative  relation  between  the  increments  in  volume, 
pressure,  and  temperature  of  a  gas  when  a  small  quantity  of 
heat  dQ  is  added  to  or  taken  away  from  it.  This  relation 
contains  the  physical  constants  jft,  Cp ,  Cv,  k,  which  must  be 
determined  by  experiment.  Experiment  telh  us  that  Cp)  just 
like  R,  is  independent  of  pressure  and  temperature.  Hence 
Cv  and  lc  are  also  independent  of  these.  This  makes  the 
thermodynamic  study  of  the  behavior  of  perfect  gases  com- 
paratively easy. 

It  is  well  now  to  consider  how  we  can  apply  the  first  law  of 
thermodynamics  to  the  solution  of  a  few  simple  problems. 


Problems. 

We  shall  consider  two  problems  in  particular : 

(1)  We  shall  determine  the  amount  of  heat  absorbed  by  a 
gas  when  it  expands  under  constant  pressure. 

(2)  We  shall  determine  the  amount  of  heat  absorbed  by 
a  gas  when  it  expands  at  constant  temperature. 

To  solve  these  equations  we  must  observe  that  a  proper 
selection  among  the  three  mathematical  statements  of  the 
first  law  of  thermodynamics  will  afford  us  considerable  mathe- 
matical advantage. 

For  the  first  problem  we  select  one  of  the  formulae,  coi 
taining  v  and  p  as  independent  variables,  and  since  p$Sb$fc~^ 


20  THERMODYNAMICS  OF 

case  is  to  be  constant,  the  term  containing  dp  becomes  zero. 
Hence 


dQ  =         —vdp  + 
becomes 


dQ  =      pdv. 

Integrating  from  v  =  vv  to  v  =  #2  ,  we  get 


Numerical  Example.  —  Consider  a  kilogramme  of  air  at 
barometric  pressure  of  760  mm.  ;  that  is,  let  the  initial  press- 
ure be  10333  kilog.  per  square  meter,  let  the  initial  volume 
be  .7733  cubic  meter,  and  let  vz  =  2vt. 

Since  cp  =.2375  kg.  calories,  and  Cp  =  .2375  X  424, 
therefore  the  amount  of  heat  absorbed  by  the  gas  while 
expanding  under  constant  pressure  from  vl  to  2vt  will  be 

_  10383  X  2376  X  424  x  ^  =  ^  fcg 


27490  . 
:  "424"  kg*  CaL 

In  the  second  problem,  where  we  wish  to  determine  the  amount 
of  heat  that  must  be  supplied  to  the  gas  in  order  to  expand 
it  under  constant  temperature,  we  take  the  formula  (la) 


REVERSIBLE  CYCLES  IN  OASES  AND    VAPORS.      21 
because  there  dT  being  zero,  when  T  is  constant,  we  obtain 


dQ  =  —  <fc,    and  hence     Q  =  RT  -  =  RT  log      , 


which  is  the  total  amount  of  heat  measured  in  kg.  meters 
which  is  absorbed  by  the  gas  while  expanding  isothermally 
from  volume  vv  to  volume  vy  To  determine  the  total  amount 
of  work  done,  we  put 

W=    I     pdv^RTI     —  =  RTlog-\ 

JVl    '  JVl    *  v> 

This  result,  compared  with  the  preceding  one,  gives  an 
identity.  It  tells  us  that  the  entire  amount  of  heat  put  into 
the  gas  was  utilized  to  do  external  ivork  ;  for  since  the  temper- 
ature remains  constant,  none  of  the  heat  supplied  goes  to  in- 
crease the  intrinsic  energy  of  the  gas. 

Isothermal  and  Adiabatic  Curves. 

If  we  make  the  supposition  that  during  the  expansion  of 
the  gas  the  temperature  remains  constant,  we  have  from  Ma- 
riotte-Gay-Lussac's  law  pv  =  O.  • 

Eepresent  this  result  graphically,  taking  p  for  ordinates  and 
v  for  abscissae.  The  resulting  curve  is  an  equilateral  hyper- 
bola referred  to  its  asymptotes.  Such  a  curve  is  called  an  iso- 
thermal curve.  (Fig.  1.) 

There  is  evidently  an  isothermal  curve  for  every  tempera- 
ture; hence  an  infinite  number  of  them,  but  everyone  of 
them  is  an  equilateral  hyperbola  in  the  case  of  a  perfect  gas. 


22 


THERMODYNAMICS  OF 


The  isothermal  curves  of  a  perfect  gas  form  a  system  of 
curves  which  never  intersect.  For  suppose  there  is  a  point 
where  two  curves  do  intersect,  this  would  have  to  be  a  point 
where  at  different  temperatures  the  pressure  is  the  same  for 
the  same  volume, — which  is  absurd. 


FIG.  1.  FIG.  2.   v 

Let  us  suppose  again  that  we  diminish  the  pressure  without 
introducing  additional  heat.  The  piston  will  go  up,  hence 
the  temperature  will  diminish;  and  as  soon  as  a  point  is 
reached  where  p^v^  =  RT^  the  expansion  will  cease. 

If  we  plot  a  curve  expressing  a  relation  between  p  and  v  at 
any  moment  during  this  expansion,  we  get  what  is  called  AN 

ADIABATIC  Or  ISENTEOPIC  CURVE.       (Fig.  2.) 

The  equation  of  this  curve  will  be  deduced  presently. 

Equation  of  the  Adiabatic  Curve  of  a  Perfect  Gas. 
Since  no  heat  is  communicated  during  an  adiabatic  expan- 
sion, the  following  relation  must  exist  at  any  moment  between 
the  physical  constants  of  the  gas: 


-  0,,)  log  £-  = 


HE  VERBID  IE  CYCLES  IN  GASES  AND    VAPORS.       23 
or 


Let  us  illustrate  this  by  an  example. 

Suppose  the  initial  temperature  is  0°  0.  above  the  freezing- 
point,  and  a  certain  initial  .  volume  of  a  kg.  of  air  is  com- 
pressed to  \  the  volume;  calculate  the  rise  in  temperature. 

T,  =  273,  ^-  =  2. 

.-.  JL  =  (2).«  =  1.329,     or     7;  =  363°  C. 

Hence     tz  =  T2  —  273  =90°  C.  above  the  freezing-point. 
Compressing  the  initial  volume  to  5,  we  find 

t  =  209°  C. 

Compressing  the  initial  volume  to  TV,  we  find 
t  =  429°  C. 

We  shall  now  calculate  the  external  work  done,  when  the  gas 
expands  adiabatically. 
From  pv  =  RTwe  get 


But  we  found  = 


p        /V  \k  n  v  fc 

Hence  ~  =  \~7l  '  OY'  more  generally,  p  =  -L^-. 


24  THERMODYNAMICS  OF 

The  last   equation  is  the  equation  of  an  adiahntic  of   a 
perfect  gas.     It  is  evidently  of  the  form 

const. 


To  calculate  the  external  work  done  during  an  adiabatic 
expansion  from  volume  v1  to  v^ ,  we  have 

w  —     I         7  k    I    *c^v        Pivik  j     1  1     ) 


^4  family  of  adiabatic  curves  never  intersect  eacli  other,  but 
every  adiabatic  curve  intersects  every  isothermal  curve. 

The  axes  of  p  and  v  are  asymptotes  to  the  adiabatics  of  a 
perfect  gas,  but  the  adiabatic  passing  through  a  given  point 
is  more«  steeply  inclined  to  the  p  axis  than  the  isothermal 
passing  through  the  same  point. 

Summary. 

It  has  been  shown  so  far  that  heat  is  a  form  of  energy,  and 
that  therefore  it  obeys  the  Principle  of  Conservation  of  Energy. 
The  constant  ratio  at  which  it  is  convertible  into  mechanical 
energy  is  424;  that  is  to  say,  if  the  quantity  of  heat  which  must 
be  supplied  to  the  unit  mass  of  the  standard  substance  (one 
cubic  decimeter  of  pure  distilled  water  at  4°  C.  and  760  mm. 
barometric  pressure)  in  order  to  raise  its  temperature  lc  C.  be 
called  a  unit  of  heat,  then  this  unit  of  heat  is  equivalent  to 
424  mechanical  units  of  work,  the  weight  of  the  unit  mass  at 


REVERSIBLE  CYCLES  IN  OASES  AND    VAPORS.       25 

a  definite  place,  that  is,  the  kilogramme  weight,  being  taken 
as  the  unit  of  force  and  the  meter  as  the  unit  of  length. 
Briefly  stated,  the  mechanical  equivalent  of  a  kilogramme 
calorie  is  424  kilogramme-meters.  After  that  it  was  shown 
that  if  we  add  a  quantity  of  heat  to  a  body  that  heat  will 
appear  partly  as  an  increment  of  the  intrinsic  energy  of  the 
body  and  partly  as  external  work.  For  very  small  quantities 
of  heat  this  can  be  stated  as  follows  : 


which  is  the  most  general  form  of  the  First  Law  of  Thermo- 
dynamics. In  this  equation  dUl  is  the  increment  of  the  in- 
trinsic energy  due  to  increase  of  the  sensible  heat  of  the  body, 
and  d  £72  is  the  increment  of  the  intrinsic  energy  due  to  the 
increment  of  internal  potential  energy  of  the  body.  The  ex- 
ternal work  dW  was  shown  then  to  be  pdv,  as  long  as  we  limit 
ourselves  to  a  particular  kind  of  external  work,  that  is,  work 
done  by  the  body  in  expanding  against  a  uniform,  normal 
surface  pressure. 

Limiting  ourselves  to  such  physical  processes  in  which  the 
state  of  the  body,  which  is  the  seat  of  these  processes,  can  be 
described  completely  at  any  moment  by  temperature,  vol- 
ume, and  pressure,  it  was  shown  that  the  first  law  of  thermo- 
dynamics can  be  stated  as  follows  : 


atr       fdi 
#+i 


The  application  of  the  first  law  of  thermodynamics  to  the 
study  of  such  physical  processes  in  various  classes  of  physical 


26  THERMODYNAMICS   OF 

bodies  was  then  in  order.  AVe  commenced  with  the  simplest 
class  of  physical  bodies,  that  is,  with  perfect  gases.  Starting 
with  the  Boyle  and  the  Mariotte-Gay-Lussac  Laws,  and  re- 
membering that  the  intrinsic  energy  of  a  perfect  gas  is  inde- 
pendent of  its  volume,  we  showed  that  the  First  Law  of  Ther- 
modynamics for  perfect  gases  can  be  stated  as  follows  : 


when  Cv  is  the  specific  heat  of  the  gas  at  constant  volume, 
measured  in  mechanical  units.  By  the  relations  Cp  =  Cv-{-  R 
and  pv  =  RT  we  then  gave  the  various  forms  of  statement  of 
the  above  equation,  these  various  statements  differing  from 
each  other  in  the  selection  of  the  independent  variables  p,  v, 
and  T,  and  the  physical  constants  Cp,  Cv,  JR,  and  k.  An 
important  observation  was  then  made,  and  that  was,  that  ac- 
cording to  Regnault's  experiments,  Cp  and  therefore  Cv,  R, 
and  Tc,  are  the  same  for  all  temperatures,  volumes,  and  press- 
ures at  which  the  gases  retain  their  characteristic  properties 
of  a  perfect  gas. 

The  application  of  these  various  forms  of  statement  of  the 
First  Law  of  Thermodynamics  for  perfect  gases  was  then  made, 
and  the  isothermal  and  adiabatic  changes  of  such  gases  dis- 
cussed. We  ended  with  the  discussion  of  isothermal  and  adia- 
batic or  isentropic  curves  of  a  perfect  gas. 

We  could,  following  the  same  method  of  discussion,  apply 
now  the  first  law  of  Thermodynamics  to  the  study  of  physical 
processes  of  above  description  in  the  case  of  any  other  class  of 
physical  bodies.  It  is  evident,  however,  that  since  the  fun- 
damental relation  between  pressure,  volume,  and  temperature 


REVERSIBLE  CYGLES  IN  GASES  AND    VAPORS.       27 

in  the  case  of  physical  bodies  in  general  is  far  from  being  as 
simple  as  in  the  case  of  perfect  gases, — it  is  evident,  we  say, 
that  our  discussion  would  lead  to  very  complicated  equations, 
involving  physical  constants  which,  in  very  many  cases,  have 
not  as  yet  been  determined  experimentally.  We  shall  there- 
fore abstain  from  a  general  discussion,  but  pass  on  to  the 
application  of  the  first  law  of  thermodynamics,  to  the  study 
of  reversible  processes  in  vapors.  This  class  of  bodies  is  ex- 
tremely important,  particularly  from  an  engineering  stand- 
point, because  it  is  the  action  of  heat  upon  these  bodies  that 
is  generally  employed  as  a  means  of  transforming  heat  energy 
into  mechanical  energy,  and  vice  versa. 

Before  making  this  next  step,  it  is  advisable  to  deduce 
another  general  law,  which,  like  the  first  law  of  thermody- 
namics, underlies  all  reversible  heat  processes  in  nature.  It 
is  called  the  Second  Law  of  Thermodynamics. 

We  close  now  this  part  of  our  course  with  a  discussion  of 
the  various  methods  of  expressing  the  specific  heats  of  gases, 
and  the  relations  which  exist  between  them. 

ON  THE  VAKIOUS  WAYS  OF  EXPRESSING  THE  SPECIFIC 
HEATS  OF  PERFECT  GASES. 

r> 

Consider  the  relation  cp  =  cv  -j — j . 

J 

This  relation  says :  Given  the  spec,  heat  of  a  gas  at  con- 
stant pressure,  the  spec,  heat  at  constant  volume  can  be 
calculated,  if  the  constant  R  of  the  gas,  and  the  mechanical 
equivalent  of  heat,  are  known. 

We   mentioned   previously  that  we   could   detfl0jSnJb4i&  ^ 


2  THERMODYNAMICS   OF 

ratio  of  the  spec,  heats  by  means  of  the  velocity  of  sound. 
This  gives  another  method  of  calculating  cv  when  cp  is 
known.  But  it  is  very  difficult  in  some  gases  to  determine  (k) 
by  velocity  measurements.  In  such  cases  we  must  take  our 
recourse  to  the  above  formula. 

R  can  be  calculated  from  the  formula 


R  - 

~ 


273' 


It  is  possible,  nowever,  that  the  gas  cannot  oe  well  observed 
at  the  freezing  temperature;  hence  we  must  look  for  another 
method  of  calculating  R. 
We  can  write  in  every  case 


At  the  same  temperature  and  pressure  we  shall  have  for  the 
same  weight  of  a  standard  gas,  say  well-dried  air, 

R'=.     Hence     R=R'-. 


v 
The  fraction  —  is  the  reciprocal  value  of  the  spec,   weight 

of  the  gas,  that  is  of  its  density,  the  density  of  air  being  taken 
is  equal  to  unity. 

Calling  this  spec,  weight,  that  is,  the  density,  d}  we  have 


KEVEliSIBLE  CYCLES  IN  GASES  AND    VAPORS.       29 
By  substituting  this  value  of  R  in  the  first  equation,  we  get 

R'  1 


_ 

r 


R'  in  our  case  is  29.27. 
Example.     Calculate  cv  for  air. 


on  cm 

d-lf     hence     cv  =  .2375  -  -~-  =  -1684. 


That  is  to  say,  to  raise  the  temperature  one  degree,  a  kilog. 
of  air  requires  only  about  \  of  the  heat  that  a  kilog.  of  stand- 
ard water  does. 

The  spec,  heats  denoted  by  cp  and  cv  refer  to  a  unit  weight 
of  the  gas,  and  their  unit  is  that  quantity  of  heat  which  is 
necessary  to  raise  the  temperature  of  a  unit  mass  of  standard 
water  under  standard  conditions  one  degree;  that  is,  in  other 
words,  the  gas  is  compared  calorically  by  weiglit,  to  water. 
While,  as  a  matter  of  fact,  it  is  sometimes  more  convenient 
to  compare  in  this  respect  gases  with  air,  by  considering 
equal  volumes,  i.e.,  to  determine  the  spec,  heat  in  such  a 
manner,  that  the  amount  of  heat  necessary  to  raise  the  unit 
volume  of  a  gas  one  degree,  is  compared  with  the  quantity 
of  heat  necessary  to  raise  a  unit  volume  of  air  through  one 
degree  (at  the  same  temperature  and  pressure). 

We  can  employ  that -method  of  comparison  for  both  specific 


30  THERMODYNAMICS   OF 

heats,  considering  in  one  case  that  the  gas  and  the  air  are 
heated  at  constant  pressure,  and  in  the  other  that  the  gas  and 
air  are  heated  at  constant  volume. 

Let  the  volume  of  a  unit  weight  of  gas  he  v. 

The  amount  of  heat  which  a  unit  volume  must  get  to  raise 
its  temperature  one  degree  at  constant  pressure  must  there- 
fore be 


Let  the  volume  of  a  unit  weight  of  air  —  v1 ,  then  the 
amount  of  heat  necessary  to  raise  a  unit  volume  through  one 
degree  under  the  same  conditions  will  be 


The  ratio  of  these  quantities  evidently  gives  us  yp;  that  is, 
the  quantity  of  heat  necessary  to  raise  the  temperature  of  the 
unit  volume  of  a  gas  at  constant  pressure  measured  in  terms 
of  the  heat  necessary  to  raise  the  temperature  of  the  unit 
volume  of  air  under  the  same  conditions.  Hence 


-  Y_    -  _fk  -.       CP 


Similarly, 


cv    , 

yv  =--,*. 


REVERSIBLE  CYCLES  IN  OASES  AND   VAPORS.       31 
TABLE   OF  SPECIFIC  HEATS  OF   GASES, 


-A 

^ 

Of 

i£ 

Density. 

Specific  Heat  at 
Const.  Pressure. 

Specific  Heat  at 
Const.  Volume. 

• 

CP 

IP 

cv 

Vu 

Air 

02~ 
Ni 
H2 
C12 
CO 
HC1 
CO, 
H2O 

1. 
1.1056 
0.9713 
0.0692 
2.4502 
0.9673 
1.2596 
1.5201 
0.6219 
0.5894 
1.5890 
2.5573 

0.2375 
0.21751 
0.24380 
3.40900 
0.12099 
0.2450 
0.1852 
0.2169 
0.4805 
0.5084 
0.4534 
0  4797 

1. 
1.013 
0.997 
0.993 
1.248 
0.998 
0.982 
1.39 
1.26 
1.26 
3.03 
5.16 

0.1684 
0.1551 
0.1727 
2.411 
0.0928 
0.1736 
0.1304 
0.172 
0  370 
0.391 
0.410 
0.453 

1. 

1.018 
0.996 
0.990 
1.350 
0.997 
0.975 
1  55 
1.36 
1.37 
3.87 
6.87 

Oxvfifcn 

Nitrogen  

Hydrogen  
Chlorine  
Curb,  monoxide.  . 
Hydroch.  ucid  gas. 
Carbonic  acid  
Steam                 .... 

Ammonia     

NH3 
C2H60 
C4H100 

Alcohol  

Ether     

III. 


SECOND  LAW  OF  THERMODYNAMICS. 

CARNOT'S  REVERSIBLE  ENGINE  AND  CARNOT'S  CYCLE. 

Let  ns  start  with  a  process  of  expansion  and  compression 
of  a  perfect  gas  enclosed  in  a  cylinder  which  is  impermeable 
to  heat.  The  piston  which  shuts  the  gas  in,  can  glide  up  and 
down  without  friction.  The  piston  being  under,  a  given  pres- 
sure, the  gas  will  at  a  given  temperature  occupy  a  definite 
volume.  Heat  can  be  communicated  to  the  gas  through  plug 
p.  Let  the  length  oe  on  the  abscissa  denote  the  original 
volume  v1  of  a  gas,  and  the  ordinate  ea  the  initial  pressure 
jo,.  Evidently  these  two  conditions  determine  the  tempera- 
ture of  the  gas  at  that  moment.  Let  the  gas  expand  isother- 


32  THERMODYNAMICS  OF 

mally.  To  accomplish  this  imagine  it  connected  to  a  large 
reservoir  A  (Fig.  4)  of  temperature  Tl ,  the  same  as  that  of  the 
gas.  By  gradually  diminishing  the  pressure  and  expanding 


/       / 
FIG.  3. 


FIG.  4. 


very  slowly,  the  temperature  will  remain  constant  and  equal 
to  Tl .  Plotting  the  curve  expressing  the  relation  between  p 
and  V)  we  get  a  part  of  an  isothermal  curve.  Point  b  indicates 
the  volume  and  pressure  at  the  end  of  the  isothermal  expan- 
sion. Now  cut  off  the  heat  reservoir  A,  and  let  the  gas  ex- 
pand slowly  still  further  (by  diminishing  the  pressure),  but 
of  course  adiabatically,  that  is  to  say,  the  gas  expands  and 
decreases  in  temperature  on  account  of  loss  of  heat  due  lo 
external  work  done.  The  curve  of  expansion,  be,  will  be  a 
part  of  an  adiabadic.  Call  the  temperature,  when  volume  vz 
has  been  reached,  T3.  From  now  on  compress  the  gas,  but 
having  first  connected  the  cylinder  to  a  reservoir  B  of  tem- 
perature T^  and  of  so  large  a  capacity  that  it  will  take  up 
all  the  heat  from  the  compressed"  gas  without  changing  its 
temperature  perceptibly.  The  heat  generated  in  the  gas  by 
compression  will  be  given  off  to  B,  hence  the  compression 
will  be  isothermal.  The  curve  of  compression  will  again  be 


REVERSIBLE  CYCLES  jtf  GA8ES  AND    VAPORS.       33 

a  part  of  an  isothermal.  At  this  point  we  impose  a  particular 
condition,  namely,  that  the  gas  be  compressed  until  a  point 
is  reached  where  the  isothermal  cuts  that  adidbatic  curve 
ivhich  goes  through  the  starting  point  a. 

Arrived  at  that  particular  point,  we  disconnect  the  reservoir 
B  and  continue  the  compression.  From  there  on  the  gas  is 
compressed  adiabatically,  the  temperature  rises,  and  when  the 
original  temperature  Tl  has  been  reached  the  gas  will  evi- 
dently have  the  same  volume  and  will  be  under  the  same 
pressure  as  at  the  beginning.  These  four  operations  con- 
stitute what  is  called  a  Carnot  Cycle. 

Now  we  shall  consider  the  work  done  by  the  gas  and  upon 
the  gas  during  the  various  operations  in  the  cycle.  The  work 
done  during  the  first  expansion  can  be  represented  by  the 
area  abfe,  that  during  the  second  by  I  cgf. 

During  the  compressions  work  has  been  spent  upon  the  gas 
equal  to  the  areas  cdhg  and  hdae  respectively.  The  difference 
between  the  work  done  and  the  work  expended  is  the  work 
gained.  This  gain  must  evidently  be  at  the  expense  of  heat 
given  up  to  the  gas  by  the  reservoir  A. 

Heat  was  taken  up  from  A  by  the  gas  during  the  first 
operation;  let  us  call  that  QlS  and  during  the  isothermal 
compression  heat  was  given  out  from  the  gas  to  the  reservoir 
j5;  let  us  call  that  §2.  Since  on  the  whole  a  certain  quantity 
Q  was  transformed  into  work  represented  by  the  area  abed,  it 
is  evident  that 


Let  us  illustrate  that  method  by  an  example, 


34  THERMODYNAMICS  OF 

Say  that  we  have  .5  kilogr.  of  air,  and  an  initial  pressure  of 
10  atmospheres,  that  is  equal  to  7600  mm.  barometric  pressure 
or  103,330  kilogr.  per  square  meter. 

Let  the  initial  volume  be       cubic  m. 


vl  =  .1  cm. 

jo,  =  103,330  klg.  per  sq.  m. 
Weight  =  .5  kg. 


103330  X  .1  X  2  0 


29.27 


_        0 

"          * 


Suppose -now  that  we  work  between  the  reservoirs  A  and  B 
at  706°  and  293°  respectively.     Hence 

Tv  =  708°,         ?;  =  293°. 

We  will  proceed  by  allowing  the  gas  to  expand  isothermally, 
until  the  volume  is  equal  to  va,  and  suppose  for  convenience 
of  calculation 

^  =  e  =  2.72. 


Work  done  by  the  gas  during  this  expansion  =    /    pdv—  W 


/** 

ion  =    /     pdv— 

*./   Vi 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.       35 
But  in  isothermal  expansion  pv  —  RTl  —  constant. 


*  1 
RT,  ^-  =  RTl  log  V-±=  29.27  X  706  kg.  meters. 

ty  vl      v  .1 


29.27  X  706  . 
Heat  taken  up  from  A  will  be  =  -  —  —  --  kg.  calories. 


Passing  to  the  second  operation  we  expand  the  gas  from 
now  on  adiabatically,  until  its  temperature  is  293°.  The 
final  volume  will  be  v3. 

Let  the  work  done  during  this  operation  be  denoted  by  PFa, 
then 


PT3  =    /  pdv. 

\s  v2 

K 

But  during  this  operation  p  =  ~-?. 


=  29,480  kg.  meters. 

Compress  now  isothermally  until  the  point  is  reached  where 
the  isothermal  cuts  the  adiabatic  which  passes  through  the 
starting  point  plvl  and  denote  this  work  of  compression  by 
W\.  Then  if  vt  be  the  final  volume, 


36  THERMODYNAMICS  OF 


(v\k~l      T  fv\k~l      T    .  . 

But  since    (  —  j      =  ~,    also     (  —  M      =  ~,  it  follows  that 

Wj/  J.  a  Wj/  -/  2 


Similarly  the  work  done  during  the  last  adiabatic  compression 
will  be 


The  total  external  work  represented  by  the  area  abed  is 
W=  W       W       W       W 


w 

Efficiency     =  •==• 


T  —  1 

•*  * 


This  relation  expresses  the  second  law  of  thermodynamics, 
but  only  in  a  very  limited  form  ;  that  is  to  say,  this  relation 
being  simply  a  mathematical  expression  for  the  efficiency  of  a 
particular  machine,  Carnot's  engine,  operating  in  a  perfectly 
definite  manner, — that  is,  by  simple  reversible  cycles,  upon  a 
particular  class  of  bodies,  namely,  perfect  gases, — this  relation 
should  not  be  considered  to  be  anything  more  than  it  really  is, 


REVERSIBLE  CYCLES  IN  OASES  AND    VAPORS.       37 

namely,  a  law  which  holds  true  for  a  particular  class  of  physi- 
cal processes  taking  place  in  a  particular  class  of  physical 
bodies. 

It  was  in  this  form  that  the  great  French  engineer,  Sadi 
Carnot,  first  discovered  the  law,  seventy  years  ago,  in  1824. 
But  he  did  more  than  that.  He  also  pointed  out  that  it  is 
only  a  particular  form  of  a  more  general  law,  that  is  to  say,  a 
law  which  holds  true  for  a  class  of  physical  processes  much 
larger  than  the  class  we  have  just  described.  It  was  by  a 
train  of  reasoning  first  suggested  by  Carnot's  great  essay, 
"Beflexions  sur  la  puissance  motrice  du  feu  et  snr  les 
machines  propres  a  la  developper,"  that  subsequent  inves- 
tigators, and  foremost  among  them  Clausius  and  Eankine, 
were  able  to  extend  the  above  relation  into  a  general  law, 
called  the  second  law  of  thermodynamics. 

Before  proceeding  any  further  in  our  discussion  towards  the 
generalization  of  the  above  efficiency  relation  into  the  second 
law  of  thermodynamics  let  us  first  examine  carefully  the 
foundation  on  which  this  relation  rests. 

.  Consider  each  one  of  the  four  operations  which  taken  to- 
gether constitute  Garnet's  cycle.  Any  one  of  them  when 
reversed  will  produce  just  the  opposite  effect.  Take  for  in- 
stance the  second  operation,  that  is,  the  first  adiabatic  expan- 
sion. If  at  the  completion  of  this  operation  we  reverse  the 
operation,  we  can  bring  the  gas  back  into  the  state  which  it 
had  at  the  beginning  of  this  operation.  The  reversed  opera- 
tion will  cost  us  mechanical  work  of  compression  equal  to  the 
area  ~bcgh.  That  work  will  appear  as  intrinsic  energy  in  the 
gas,  that  is,  as  heat,  in  consequence  of  which  the  temperature 
of  the  gas  is  increased  from  T%  to  T^  If  the  first  operation  be 


38  THERMODYNAMICS  OF 

called  the  direct  and  the  second  the  reversed,  then  we  can  say 
that  the  direct  and  the  reversed  operation  just  neutralize  each 
other.  The  same  is  true  of  every  other  operation  of  the  cycle. 
It  follows,  therefore,  that  the  whole  cycle  can  be  reversed  and 
the  reversed  cycle  will  produce  just  the  opposite  effect  of  the 
direct  cycle,  that  is  to  say,  it  will  deprive  the  reservoir  B  of 
Q^  units  and  add  Q1  units  of  heat  to  reservoir  A,  and  it  will 
also  transform  Ql  —  Q^  =  Q  units  of /work  into  heajj  In  the 
first  cycle  heat  passes  from  a  hot  bo^jToTaTcold  body,  but 
some  of  this  heat  in  its  journey  to  the  cold  body  is  transformed 
into  mechanical  work  and  therefore  never  reaches  the  cold 
body.  In  the  reversed  cycle  mechanical  work  is  expended,  in 
order  to  transfer  §2  units  of  heat  from  the  cold  body  to  the 
hot,  and  in  addition  generate  Q  units  of  heat  which  are  also 
given  up  to  the  hot  body. 

Such  a  cycle  is  called  a  reversible  cycle.    The  relation 

Q  _  r.  -  T, 
<?,"      r, 

is  evidently  the  same  no  matter  whether  the  cycle  be  taken 
in  the  direct  or  in  the  reversed  sense. 

Suppose  now  that  during  any  one  of  the  operations  the  gas 
had  changed  chemically,  but  in  such  a  way  that  its  chemical 
constitution  could  not  be  restored  by  reversing  the  operation. 
Such  an  operation  would  not  be  reversible,  nor  would  the 
cycle  of  which  this  operation  forms  a  component  part  be  re- 
versible. In  such  a  cycle  the  above  relation  may  or  may  not 
be  true.  A  special  investigation  is  required  to  clear  up  this 
point;  but  since  we  are  not  going  to  discuss  it  in  our  course 
it  is  well  that  you  should  be  told  that  it  is  not  true. 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.       39 

It  should  also  be  observed  that  there  are  a  great  many 
reversible  cyclic  processes  in  all  branches  of  physical  sciences 
which  do  not  apparently  resemble  Caruot's  cycles  of  opera- 
tions. A  great  many  illustrations  may  be  cited  from  chem- 
istry, electricity  and  magnetism,  etc.  Whether  in  these 
Carnot's  law  of  efficiency  or  the  general  form  of  it  which  we 
are  about  to  deduce  is  applicable  or  not,  must  be  decided  by 
investigations  which  are  outside  of  the  limits  of  our  course. 
What  we  propose  to  do  is  simply  this  :  we  are  going  to  prove 
the  applicability  of  Carnot's  law  of  efficiency,  and  therefore 
of  all  its  extensions,  to  all  physical  processes  which  consist  of 
operations  any  one  of  which  can  be  performed  reversibly  by 
a  Carnot  engine. 

The  first  step  in  our  work  will  be  to  prove  that  the  efficiency 
of  a  Carnot  cycle  depends  on  the  temperatures  of  the  reser- 
voirs A  and  B  as  expressed  above,  no  matter  what  the  sub- 
stance may  be  upon  which  we  operate.  To  do  this  we  shall 
start  from  an  axiom  first  stated  by  Clansius,  and  therefore 
called  the 

AXIOM  OF  OLAUSIUS. 
Heat  cannot  pass  from  a  cold  body  to  a  hot  body  of  itself. 

This  statement  includes,  that  by  conduction  of  heat,  or  by 
concentration,  reflection,  etc.,  of  radiant  heat,  no  heat  can  be 
transferred  to  a  body  at  the  expense  of  a  colder  one,  unless 
the  process  of  transference  involves  the  expenditure  of  some 
sort  of  energy. 

In  the  preceding  discussion  we  saw  that  heat  passed  from  a 
cold  to  n  hot  reservoir,  but  not  without  expenditure  of  work. 


40  THERMODYNAMICS  OF 

So  we  might  say,  instead  of  "  of  itself,"  "  without  work  or 
compensation." 

The  truth  of  this  axiom  is  proved  in  the  same  way  as  the 
truth  of  Newton's  axioms  of  motion,  that  is,  by  an  appeal  to 
experience. 

We  shall  now  prove  that  the  ratio  between  the  heat  trans- 
formed into  work  by  a  simple  reversible  cycle  and  the  heat 
transferred  from  a  hot  body  to  a  colder  body  during  the  same 
cycle  is  independent  of  the  material  which  we  employ  in  the 
cylinder  of  our  ideal  engine,  and  that  it  can  depend  only  on 
the  temperatures  of  the  reservoirs  A  and  B. 

For  this  purpose  we  employ  two  equal  cylinders  A  and  B 
containing  two  different  substances,  upon  which  we  propose 
to  perform  simple  reversible  cycles.  Each  cycle  is  performed 
betiveen  the  same  two  reservoirs,  so  that  the  temperatures  be- 
tween which  we  operate  are  the  same  for  each  cycle.  Perform 
now  cycle  I  with  the  cylinder  A,  and  suppose  that  a  quantity 
of  heat  equal  to  Q  is  transformed  into  mechanical  work,  and 
that  a  quantity  of  heat  equal  to  ()2  is  transferred  to  the  colder 
reservoir.  Pass  now  to  the  other  cylinder  and  perform  a 
similar  cycle,  but  in  such  a  way  that  again  a  quantity  Q  of 
heat  is  transformed  into  mechanical  work.  A  certain  quantity 
of  heat,  say  ()/,  will  be  transferred  to  the  colder  reservoir. 
We  wish  to  prove  now  that  QJ  =  Q^.  For  if  it  is  not,  sup- 
pose that  §/  >  Qf  After  performing  cycle  I  in  the  direct 
order,  by  which  a  quantity  of  heat  Q  is  transformed  into 
mechanical  work  and  $2  transferred  to  the  colder  reservoir, 
employ  cylinder  B  to  perform  cycle  II  in  the  reverse  order, 
so  that  the  mechanical  work  obtained  by  cycle  I  is  retrans- 
formed  into  heat,  and  a  quantity  of  heat  equal  to  QJ  is  trans- 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.       41 

f erred  from  the  cold  to  the  hot  reservoir.  At  the  end  of  the 
two  cycles  the  only  change  that  still  remains  is  the  transfer 
of  a  quantity  of  heat  equal  to  Q9'  —  Q^  from  the  cold  to  the 
hot  reservoir,  that  is,  a  transference  of  heat  from  a  body  of 
lower  to  a  body  of  higher  temperature  without  any  compensa- 
tion. This  is  contrary  to  the  axiom  of  Clansins,  hence  the 
supposition  that  Q9'  >  §2  must  be  dismissed. 

Next  suppose  that  Qy'  <  Q9.  We  can  prove  that  this  sup- 
position also  violates  our  axiom  by  going  through  cycle  II  in 
the  direct  and  cycle  I  in  the  reverse  order. 

Hence  Olausius'  Axiom  leads  to  the  conclusion  that  QJ  = 
Q9.  That  is  to  say,  ivhencver  mechanical  work  is  obtained  at 
/he  expense  of  heat  ~by  a  reversible  cyclic  operation,  heat  is 
transferred  from  a  hot  to  a  cold  body  and  the  ratio  of  the  work 
obtained  to  the  quantity  of  heat  transferred  from  the  hot  to  fhe 
cold  body  is  independent  of  the  substance  which  is  operated 
upon. 

That  ratio  will  therefore  be  the  same  for  a  perfect  gas  as 
for  any  other  substance.  We  proceed  to  calculate  it  for  a 
perfect  gas.  Let  the  temperatures  of  the  two  reservoirs  be 
7\  and  7\  and  suppose  that  7\  >  7*a. 

We  found  previously  that  in  the  case  of  a  perfect  gas 


or 


Q  "         Q  Q  "  T,  -  T,> 


42  THERMODYNAMICS  OF 

We  see  therefore  that  this  ratio  in  the  case  of  a  perfect  ga 
and  therefore  also  in  the  case  of  any  other  substance  depend 
on  the  two  temperatures  between  which  the  cycle  is  performed 

A  slight  modification  of  this  relation  will  give  us  anothe 
form  of.  stating  the  second  law  of  thermodynamics  applicabl 
to  simple  reversible  cycles.  From  this  relation  we  obtain 


Q 


Tu       Tl  —  T^ 

We  obtained  previously 


Q 

hence 


Ql  _ 

T  ~ 


Combining  this  with  the  above  we  obtain 

£=«.,    or    f?-a  =  ft    or    |'  +  ^  =  0. 

*1  ^3  -*!  ^3  -^1  -*3 

—  Q^  is  the  heat  given  off  to  the  cold  reservoir  ~by  the  sul 
stance  operated  upon,  hence  —  (—  QJ  or  +  Q*  can  be  said  t 
be  the  heat  received  by  the  substance  from  the  cold  reservoir 
With  that  mental  reservation  we  can  write  down 


T, 


REVERSIBLE  CYCLES  IN  OASES  AND   VAPORS.       43 

This  is  the  more  compact  and  at  the  same  time  more  flexible 
form  of  stating  the  second  law  of  thermodynamics  applicable 
to  simple  reversible  cycles.  Tho  English  translation  of  this 
mathematical  statement  is  :  If  in  a  simple,  reversible,  cycle  the 
heats  received  by  the  substance  operated  upon  be  divided  by  the 
temperatures  at  which  these  heats  were  received  and  the  quo- 
tients thus  obtained  be  added  together,  then  will  the  sum  be 
equal  to  zero. 

This  statement  of  the  second  law  of  a  simple  reversible 
cycle  is  certainly  not  nearly  as  clear  nor  physically  as  in- 
telligible as  the  other  statement,  which  involved  the  efficiency 
of  an  ideal  reversible  engine,  but,  as  will  be  presently  seen,  it 
enables  us  to  extend  in  a  very  simple  manner  the  applicability 
of  this  law  to  complex  reversible  cycles  and  thus  give  the 
second  law  of  thermodynamics  a  much  more  general  form. 

GENERAL  FORM  OF  THE  SECOND  LAW  OF  THERMO- 
DYNAMICS. 

Complex  Cycles. 

So  far  we  have  considered  cycles  between  two  temperatures 
only.  We  now  proceed  to  extend  this  to  a  so-called  complex 
cycle,  that  is,  a  cycle  of  operations  between  several  reservoirs 
of  different  temperatures.  We  commence  with  three  reser- 
voirs of  temperature  Tl9  T^,  and  T3. 

Let  ab  (Fig.  5)  denote  an  isothermal  expansion  at  the  tem- 
perature Tl ,  be  an  adiabatic  expansion  down  to  the  tempera- 
ture T^ ,  cd  an  isothermal  expansion  at  the  temperature  T^ , 
de  an  adiabatic  expansion  down  to  the  temperature  T3,  e/an 
isothermal  compression  at  the  temperature  Ta ,  the  isothermal 


44 


THERMODYNAMICS  0V 


compression  to  continue  until  the  isothermal  curve  ef  cuts 
the  adiabatic  through  a,  and  lastly  fa  an  adiabatic  compres- 
sion to  the  starting  point. 

During  the  expansions  ab  and  cd  the  substance  operated 
upon  takes  up  the  quantities  of  heat  Qt ,  Q2  from  the  reser- 
voirs A  and  B  at  temperatures  Tt  and  Tt,  and  during  the 


FIG.  5. 


isothermal  compression  ef  it  takes  up  the  negative  quantity 
of  heat  —  Q3  from  the  reservoir  C  at  temperature  T3.  The 
external  work  done  during  the  cycle  is  represented  by  the 
area  abcdefa. 

We  proceed  to  show  now  that  a  similar  relation  holds  true 
in  this  case  as  in  the  simple  cycle. 

To  do  that,  let  us  produce  the  adiabatic  be  until  it  cuts  the 
isothermal  fe  at  g ;  then  will  the  whole  complex  cycle  be 
divided  into  two  simple  cycles  dbg  fa  and  cdegc. 

The  negative  quantity  Qa  consists  of  q9  and  q/ ,  given  off  to 
the  coldest  reservoir  during  the  isothermal  compressions  gf 
and  eg  respectively.  We  can  now  write  down  the  symbolical 


REVERSIBLE  CYCLES  IN  GASES  AXL    VAPORS.       45 

statement  of  the  second  law  for  each  one  of  these  two  simple 
cycles,  viz. : 


•  +  -TF  =  0  for  cycle  dbcgfa ; 


•7FT  +  %  =  °    "         "       C%C. 

Adding  these  two  we  obtain 


or,  since  g,  +  #/  = 


rp 
-*  i 


In  the  same  manner  we  can  prove  this  relation  for  a  com- 
plex cycle  divisible  into  any  number  of  simple  cycles  and 
obtain 


If  the  cycle  be  of  infinite  complexity,  that  is  to  say,  if  it  be 
representable  by  an  infinite  number  of  infinitely  small  elements 
of  aduibatic  and  isothermal  curves,  it  is  evident  that  the 

relation  ^~,  =-.  0  will  still  hold  true.     Of  course  each  one  of 


46  THERMODYNAMICS  OF 

the  Q's  will  be  infinitely  small  and  the  sum  will  be  an  in- 
finite sum.  We  can  therefore  express  this  relation  in  the 
notation  of  the  infinitesimal  calculus,  thus : 


A  reversible  cycle  of  this  complexity  will  evidently  in  tne 
limit  be  representable  by  a  continuous  closed  curve.  The 
area  bounded  by  this  curve  will  represent  the  external  work 
done  during  the  cycle  if  the  cycle  be  performed  in  the  direct 
sense,  otherwise  it  will  represent  the  external  work  spent  in 
order  to  transfer  a  certain  amount  of  heat  to  the  hot  reservoir, 
a  part  of  which  comes  from  the  cold  reservoir. 

The  statement  of  the  second  law  of  thermodynamics  (appli- 
cable to  reversible  cycles)  as  given  by  the  last  equation,  though 
very  general,  gives  us  but  little  information  about  the  various 
elements  of  which  the  cycle  is  composed.  It  gives  us  no  in- 
formation about  the  progress  of  the  various  physical  processes 
which  taken  together  constitute  the  cycle.  It  has  the  defect 
which  is  common  to  all  integral  laws.  For  such  laws  give  us 
the  total  effect  of  a  series  of  physical  processes  without  giving 
us  the  separate  effect  of  each  process  of  the  series.  Take  for 
instance  the  integral  laws  of  Keppler  which  describe  the 
motion  of  the  planets  around  the  sun.  They  tell  us  all  about 
the  areas  traced  out  by  radii  vectores  of  the  planets  during  a 
finite  time,  about  the  periodic  times,  and  about  the  orbits  of 
the  planets,  but  they  do  not  tell  us  anything  about  the  action 
which  is  going  on  at  any  and  every  moment  between  the  sun 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.       47 

and  the  planets  to  produce  these  integral  effects.  It  took  no 
less  a  genius  than  that  of  a  Newton  to  infer  these  differential 
effects  from  the  integral  effects  discovered  by  Keppler.  The 
inference  was  the  law  of  gravitation.  To  draw  that  inference 
Newton  had  to  invent  the  infinitesimal  calculus.  We  may 
well  assert  that  he  had  to  invent  it ;  for  a  simple  consideration 
will  convince  you  that  the  process  of  passing  from  an  integral 
physical  law  to  the  differential  law — that  is,  the  law  which  de- 
scribes a  physical  process  as  it  progresses  from  point  to  point 
through  infinitely  small  intervals  of  space  and  time — is  exactly 
the  same  as  the  mathematical  process  of  passing  from  an  in- 
tegral to  the  differential  of  that  integral. 

We  are  now  ready  todiscuss  the  differential  of  the  integral 


T 


Consider  any  one  of  the  infinitely  small  terms  -^  in  the 

-*    10 


FIG.  6. 

infinite  series  j  ~.    The  subscripts  12  (Fig.  6)  denote  that 
dQlz  is  the  heat  taken  up  during  the  infinitely  small  element 


48  THERMODYNAMICS   OF 

12  of  the  cycle  and  that  Tlt  was  the  mean  temperature  of  the 
substance  operated  upon  at  that  particular  moment.  It  is  evi- 
dent that  clQiy  depends  on  the  instantaneous  state  of  the  sub- 
stance operated  upon  at  that  moment,  and  also  upon  the 
nature  of  the  operation  represented  by  the  element  12.  In 
other  words,  dQ^  depends  on  the  volume,  pressure,  and  tem- 
perature, and  the  method  of  their  variation  during  the  opera- 
tion represented  by  the  element  12.  Now  all  these  things  are 
completely  defined  by  the  co-ordinates  of  the  extremities  of  the 
element,  hence  dQ^  is  a  function  of  these  co-ordinates.  The 

same  is  true  of  the  infinitely  small  quantity  -~2. 

'  12 

Consider  now  a  quantity  0,  and  suppose  it  to  be  a  finite, 
continuous,  and  singly-valued  function  of  the  pressure, 
volume,  and  temperature  of  the  substance  operated  upon. 
<p  will  therefore  have  definite  values  at  every  point  of  any 
cyclic  diagram. 

Denote  by  0, ,  02 ,  08 ,  .  .  .  0»  the  values  of  0  at  the 
various  points  1,  2,  3,  ...  n  all  around  the  cyclic  diagram 
and  consider  the  differences. 

0,  -  0,  =  40; 

03  -  0,  =  ^20; 

04  -  03  =  A0; 


01  —  0n  =  ^n0- 

It  is  evident  that  for  every  cyclic  diagram 

40  +  40  +  .  .  .  +  40  =  0. 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.       49 

When  the  points  1,  2,  3,  ...  n  are  infinitely  near  each 
other,  then  ^,0,  d^<p,  etc.,  are  infinitely  small  quantities  or 
what  we  call  in  calculus  the  complete  differentials  of  0  at  the 
various  points  of  the  curve.  We  express  that  by  saying  that 
in  the  limit  we  shall  have  the  differences  djp,  t?20,  f/30,  etc., 
but  an  infinite  number  of  them  and  their  sum,  which  will 
still  be  zero  for  every  cycle,  can  be  expressed  in  the  ordinary 
way,  thus: 


= o. 


This  relation  will  hold  true  for  every  function  of  pressure, 
volume,  and  temperature  which  has  definite  values  at  every 
point  of  the  cyclic  diagram  and  varies  continuously  from 
point  to  point  of  the  diagram.  There  are  evidently  an  in- 
finite number  of  such  functions. 

If  that  integral  is  extended  not  all  around  the  cyclic  dia- 
gram, but  only  between  two  points,  say  points  1  and  <x,  then 
we  shall  have  the  integral  evidently  equal  to  0a  —  0j  ;  that 
is  to  say, 


=  0.- 


The  value  of  the  integral  will  depend  therefore  on  the 
position  of  the  initial  and  final  point  of  integration  and  not 
on  the  shape  of  the  cyclic  diagram  between  the  two  points. 

This  relation  throws  much  light  upon  our  integral    C d<p  =  0. 


50 


THERMODYNAMICS   OF 


For,  consider  two  integrals  j  d<p  along  a   portion  AB  of  a 
cyclic  diagram  (Fig.  7);  one  integral  from  A  to  B  along  one 


FIG.  7. 


part  of  the  diagram  in  the  clockwise  direction,  and  the  other 
along  the  other  part  of  the  diagram  in  anti-clockwise  direc- 
tion. Denote  the  first  by  subscript  (AB)l ,  the  second  by  sub- 
script (AB\.  Then 


/  d(/>  =  <f>B  -  <{>A     and       /  d$  =  <j>B  — 

J(AB)l  J(AB)^ 


<t>A> 


Denote  now  the  integral  from  B  to  A  along  the  second  part 
of  the  cycle  by  subscript  (BA);  then  evidently 


(BA) 


(AB)9 


REVEHS1BLE  CYCLES  IN  GASES  AND    VAPORS.       51 

It  follows  therefore  that 

/W-f    /d0=        defy-   I  d<t>=((f>B-<f>A)-(<f>B-<f>A)=0, 

J (AB)j.          J (BA)  c/UB)i          c/(AB)o 


as  it  should  be,  since 

-j-    /  dcf)  —  integral  all  around  the  cycle. 


/  dcf)  —  i 

J(BA) 


We  can  therefore  say  that  the  integral  all  around  the  cyclic 
diagram  vanishes  because  the  value  of  the  integral  between 
any  two  points  of  the  diagram  depends  on  the  position  of 
the  two  points  and  not  on  the  shape  of  the  diagram  between 
these  two  points.  The  only  other  way  in  which  that  integral 
could  vanish  would  be  that  dcp  is  zero  for  every  element  of 
every  cycle,  in  which  case  0  would  be  a  constant.  This  is 
contrary  to  our  supposition,  for  we  supposed  that  it  is  a 
variable  function  of  volume,  pressure,  and  temperature. 
A  careful  inspection  of  the  series  which  in  the  limit  gave 

the  integral  J  d$  will  convince  you  that  this  integral  be- 
tween any  two  points  of  any  cyclic  diagram  depends  on 
the  position  of  these  two  points  and  on  nothing  else,  because 
the  quantity  under  the  integral  sign  is  a  perfect  differen- 
tial of  a  function  0  which  is  finite,  continuous,  and  singly 
valued  at  every  point  of  any  cyclic  diagram.  We  conclude, 

therefore,  since  the  integral    i -^  taken  around  a  diagram 


52  THERMODYNAMICS   OF 

representing  any  reversible  cyclic  process  vanishes,  that  -~ 

is  the  complete  differential  of  a  function  S  which  is  a  finite, 
continuous,  and  singly-valued  function  of  the  volume,  press- 
ure, and  temperature  of  the  substance  upon  which  we  oper- 
ate. We  express  this  by  writing 


This  is  the  most  general  form  of  the  second  laiv  of  thermo- 
dynamics. This  statement  of  the  second  law  bears  about  the 
same  relation  to  the  other  statement, 


as  the  Newtonian  law  of  universal  gravitation  bears  to  the 
three  laws  of  Keppler  which  describe  the  motion  of  planets 
around  the  sun.  It  tells  us  all  about  the  progress  of  the 
infinite  number  of  infinitely  small  processes  which,  taken 
together,  constitute  the  whole  reversible  cyclic  process  of 
which  the  cyclic  diagram  is  a  geometrical  representation. 
We  shall  illustrate  this  statement  by  reference  to  a  par- 
ticular physical  process  when  we  come  to  the  discussion  of 
the  application  of  the  two  laws  of  thermodynamics  to  the 
study  of  saturated  vapors,  which  we  are  now  ready  to  take  up. 
This  statement  of  the  second  Liw  of  thermodynamics  of 
reversible  processes  was  first  given  by  Clausius.  The  func- 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.       53 

tion  S  was  named  by  him  the  Entropy  of  the  system.  It 
is  impossible  to  give  a  general  and  yet  complete  definition 
of  the  physical  meaning  of  the  entropy  function,  because 
this  meaning  will  vary  considerably  with  the  nature  of  the 
process  that  we  may  be  considering.  Hence  it  can  be  dis- 
cussed only  in  connection  with  the  discussion  of  each  par- 
ticular process  under  consideration.  It  is  desirable,  however, 
to  illustrate  its  meaning  by  considering  a  few  simple  pro- 
cesses. 

1st.  Suppose  a  body  expands  reversibly  at  constant  tem- 
perature from  volume  vl  and  pressure  pl  to  volume  v^  and 
pressure  pz.  Let  the  heat  that  must  be  supplied  to  the 
body  be  Q,  then 

Q=T(S,-$l); 

i 

that  is,  the  heat  is  equal  to  the  absolute  temperature  multi- 
plied by  the  increase  of  the  entropy.  In  the  case  of  a 
perfect  gas  that  heat  is  also  equal  to  the  external  work 

done   by  the  gas  during  the  expansion,  and   this  work  we 

v 
found  to  be  equal  to  RT  log  — . 

Hence  for  isothermal  expansions  of  perfect  gases  we  have 


2d.  If  the  gas  expands  adiabatically,  then,  since  no  heat  is 
supplied  during  such  expansion,  we  shall  have 

0  =  TdS,     hence    dS  =  0, 


54  THERMODYNAMICS  OF 

That  is  to  say,  the  entropy  remains  constant.  It  is  on 
this  account  that  adiabatic  expansion  is  called  sometimes  an 
isentropic  expansion,  the  word  isentropic  meaning  that  the 
entropy  remains  constant. 


IV. 


APPLICATION  OF  THE  TWO   LAWS  OF  THERMODY- 
NAMICS TO   SATURATED  VAPORS.  

We  are  now  ready  to  take  up  the  discussion  of  the  rever- 
sible processes  in  the  next  simplest  class  of  physical  bodies, 
that  is,  saturated  vapors.  Our  discussion  will  be  framed 
after  the  model  of  our  discourse  on  the  reversible  cyclic  pro- 
cesses in  the  case  of  perfect  gases.  We  shall  therefore  com- 
•mence  with  an  appeal  to  experience  for  information  about 
the  physical  properties  of  sat  united  vapors.  We  did  the 
same  thing  in  the  case  of  perfect  gases  by  taking  account  of 
the  laws  of  Boyle  and  Mariotte-Gay-Lussac,  and  of  the 
experimental  researches  of  Regnault  on  the  specific  heats  of 
perfect  gases.  We  shall  then  put  the  two  laws  of  thermo- 
dynamics into  a  suitable  form  by  taking  proper  account  of 
this  information  about  the  physical  properties  of  saturated 
vapors.  This  will  give  us  two  mathematical  statements  of 
these  laws  involving  certain  physical  constants  of  saturated 
vapors.  Some  of  these  constants  are  known  from  experi- 
mental researches  just  as  in  the  case  of  perfect  gases;  others 
we  shall  have  to  calculate  with  the  assistance  of  the  two  laws. 
Hence  after  modelling  properly  our  two  fundamental  laws, 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.       55 


we  shall  proceed,  just  as  we  did  in  the  case  of  perfect  gases, 
to  calculate  and  tabulate  the  various  physical  constants  of 
saturated  vapors.  Having  done  that,  we  shall  then  be  ready 
to  pass  on  to  the  discussion  of  isothermal  and  adiabatic 
expansions  of  saturated  vapors,  calculating  in  each  case  the 
external  work  done  and  the  heat  supplied  to  the  expanding 
vapor.  Since  water  vapor  is  the  substance  most  generally 
employed  in  our  vapor-engines,  it  is  advisable  to  concentrate 
our  attention  as  much  as  possible  upon  that  vapor.  The 
discussion  will  nevertheless  lose  very  little  of  its  generality, 
since  in  dealing  with  saturated  water-vapor  we  have  to 
consider  the  same  questions  that  would  have  to  be  con- 
sidered in  any  other  vapor. 

PHYSICAL  PROPERTIES  OF  SATURATED  VAPORS. 

Consider  a  cylinder  A  (Fig.  8)  in  which  a  piston  B  can  s^de 
np  and  down  without  friction.     Let  the  volume  enclosed  by 


Pro.  8. 


the  piston  be  denoted  by  v,  and  let  a  part  a  of  this  volume 
contain  a  volatile  liquid.     The  other  part  b  will  be  vapor  of 


56  THERMODYNAMICS  OF 

that  liquid.  Let  the  temperature  be  uniform  throughout 
and  equal  to  T.  Experience  tells  us  that  this  vapor,  as  long 
as  it  is  in  contact  ivitli  its  own  liquid,  lias  a  definite  tension 
which  depends  on  the  nature  of  the  liquid  and  on  its  tempera- 
ture and  on  nothing  else.  Let  the  pressure  on  the  piston  be 
just  such  as  to  be  in  equilibrium  with  the  tension  of  the 
vapor  at  the  temperature  T.  Suppose  now  that  the  piston  is 
pushed  down  through  a  very  small  distance ;  the  volume  of 
the  vapor  is  diminished ;  experience  tells  us  that  the  vapor 
will  condense.  Vice  versd,  if  the  piston  is  slightly  raised, 
keeping  the  temperature  constant,  then  some  more  liquid 
will  evaporate  so  as  to  keep  the  density  in  the  vapor  volume 
the  same  as  before.  We  infer,  therefore,  that  vapor  when  in 
contact  with  its  own  liquid  has  the  maximum  density  for  that 
temperature ;  in  other  words,  it  is  saturated. 

Both  the  tension  and  the  density  of  the  saturated  vapor 
increase  with  temperature. 

I  Saturated  vapors  do  not  obey  the  Mariotte-Gay-Lussac  law, 
'not  even  approximately,  except  for  low  temperatures.  So,  for 
instance,  this  law  cannot  be  applied  to  saturated  water-va- 
pors at  any  temperature  which  is  more  than  only  a  very  few 
degrees  centigrade  above  the  freezing-point. 

When  a  liquid  is  brought  into  a  space  which  does  not  con- 
tain the  vapor  of  this  liquid,  then  the  liquid  will  evaporate 
until  its  vapor  has  the  maximum  density  in  every  part  of  the 
surrounding  space.  The  process  of  evaporation  will  be  slow 
if  the  space  contains  a  gas  at  considerable  pressure,  but 
rapid  if  the  space  is  a  vacuum.  This  is  due  to  the  slowness 
with  which  the  vapor  diffuses  through  a  gas  of  considerable 
tension.  Experience  tells  us  that  evupqration  is  accompanied 


REVERSIBLE  CYCLES  Itf  OASES  AND    VAPORS.      57 

by  a  cooling  off  of  the  liquid,  and  the  cooling  is  the  more  rapid 
the  quicker  the  process  of  evaporation.  You  are  acquainted 
with  the  well-known  experiment  of  freezing  mercury  by 
bringing  it  in  contact  with  a  volatile  liquid  which  is  made  to 
evaporate  very  rapidly  by  the  action  of  a  pump.  If  we  wish 
to  maintain  the  temperature  of  an  evaporating  liquid  constant, 
then  heat  must  be  supplied  to  it.  Conversely,  if  the  saturated 
vapor  be  compressed  ffireontaet-w4th^-Us~owft-4i^«i^;  heat  will 
be  developed,  and  if  we  wish  to  keep  the  temperature  constant, 
then  this  heat  must  be  taken  away.  The  quantity  of  heat 
which  is  developed  by  compressing  a  kilogramme  of  saturated 
vapor  to  liquid  at  the  same  temperature  and  pressure  is  called 
the  latent  heat  of  saturated  vapor  at  that  temperature.  This 
is  a  very  important  physical  constant  and  has  been  deter- 
mined experimentally  for  a  large  number  of  vapors. 

When  the  temperature  of  a  liquid  has  reached  that  point  at 
which  the  tension  of  its  vapor  is  the  same  as  the  pressure  of 
the  surrounding  space,  then  the  vaporization  will  be  rapid, 
because  it  will  take  place  not  only  on  the  surface  of  the  liquid, 
but  also  in  the  interior  parts,  especially  if  the  liquid,  as  is 
usually  the  case,  contains  small  bubbles  of  a  gas,  like  air,  dis- 
solved in  it.  When  this  takes  place  we  say  that  the  liquid  is 
boiling.  The  boiling  point  of  pure  distilled  water  at  760  mm. 
barometric  pressure  is  100°  C.,  and  it  rises  with  the  rise  of 
barometric  pressure.  The  boiling  point  of  water  is  raised  by 
the  solution  of  foreign  substances  in  it.  Sea-water,  for  in- 
stance, has  a  considerably  higher  boiling  point  than  pure  dis- 
tilled water  at  the  same  barometric  pressure.  As  long  as  the 
boiling  process  continues,  the  temperature  of  the  liquid  re- 


58  THERMODYNAMICS  Off 

maiws  constant;  the  heat  supplied  to  the  liquid  is  all  utilized 
to  transform  the  liquid  into  steam. 

If,  however,  the  liquid  does  not  contain  any  dissolved  gases, 
then  the  process  of  boiling  is  considerably  retarded.  There  is 
superheating  and  the  boiling  is  accompanied  by  the  well- 
known  "  bumping  "  or  concussive  boiling. 

When  the  contact  of  a  saturated  vapor  with  its  own  liquid 
is  cut  off  and  the  vapor  allowed  to  expand  slowly,  it  will  cool 
off,  of  course,  and  its  density  will  diminish.  The  ratio  of  the 
diminution  of  the  density  to  the  diminution  of  temperature 
is  different  for  different  vapors.  In  some  vapors  the  tem- 
perature diminishes  more  rapidly  than  the  density.  Such 
vapors  will  condense  during  adiabatic  expansion.  If,  how- 
ever, the  temperature  diminishes  more  slowly  than  the  dens- 
ity, then  the  vapor  will  be  superheated,  that  is  to  say,  it  will 
not  remain  saturated  during  adiabatic  expansion.  In  the 
first  case  we  have  superheating  and  in  the  second  we  have 
condensation  during  adiabatic  compression. 


OK  THE  PHYSICAL  CONSTANTS  OF  SATURATED  VAPORS. 

Let  the  total  weight  of  the  liquid  and  its  saturated  vapor 
ill  the  above-mentioned  cylinder  be  M.  Denote  by  T  the 
absolute  temperature  of  the  whole  mass,  and  by  #the  volume. 

Let  m  =  weight  of  the  vapor  in  kilogrammes; 

p  =  tension  of  the  saturated  vapor  in  kgr.  per  sq.  m. ; 
a  =  specific  volume  of  the  liquid; 
s  =         "  «      ><     «    vapor. 


REVERSIBLE  CYCLES  IN  OASES  AND    VAPORS.       59 
We  shall  have  then 

ms  +  (M  —  m)<r  =  v, 
or 

m(s  —  cr)  -f-  Mcr  =  v. 

In  what  follows  we  shall  write  u  for  s  —  cr,  hence 
mu  -\-  Mcr  =  v. 

It  must  be  observed,  however,  that  (in  the  case  of  water  es- 
pecially) s  is  large  in  comparison  to  cr,  so  that  in  most  cases 
we  may  consider  u  practically  equal  to  s. 

When  the  temperature  77and  the  weight  m  of  the  saturated 
vapor  are  given,  then  the  state  of  the  whole  mass  in  the  cylin- 
der is  completely  known.  For  we  know  then  the  pressure 
and  the  volume  (since  the  density  is  completely  determined 
by  T),  and  that  is  all  that  we  require  to  know.  We  can  there- 
fore select  T  and  m  as  our  variables. 

In  order  to  bring  out  clearty  the  meaning  of  some  other 
physical  constants  of  saturated  vapors,  we  shall  consider  now 
two  distinct  physical  processes  which  we  shall  call  definitional 
processes. 

DEFINITIONAL  PROCESS  (A). 

If  a  small  quantity  of  heat  dQ  is  communicated  to  the  masg 
without  changing  the  weight  in  of  the  saturated  vapor,  we  §hall 
have  the  following  changes  produced; 


60  THERMODYNAMICS  OF 

First.  The  temperature  of  the  liquid  will  increase  by  a  small 
quantity  dT.  Hence  if  C  denote  the  specific  heat  of  the 
liquid  at  the  temperature  T  (measured  in  kg.  meters),  then 
(M '  —  m)CdTwi\l  be  equal  to  that  part  of  dQ  which  is  used 
up  in  raising  the  temperature  of  the  liquid  by  dT. 

Secondly.  The  temperature  of  the  vapor  is  increased 
by  the  same  amount  dT.  Hence  its  density  will  increase; 
and  if  we  did  not  diminish  the  volume  slightly  by 
pushing  down  the  piston,  the  weight  m  of  "the  vapor 
part  of  our  whole  mass  would  increase.  Supposing  now 
that  by  a  slight  depression  we  keep  m  constant,  there  will 
be  no  vaporization  and  the  rest  of  dQ  will  go  to  increase 
simply  the  temperature  of  the  m  kg.  of  the  saturated  vapor 
by  dT  degrees. 

This  point  needs  a  little  fuller  discussion  on  account  of  its 
importance  in  the  theory  of  the  steam-engine.  Suppose  that 
we  have  1  kilogramme  of  vapor  in  saturated  state  but  not  in 
contact  with  its  own  liquid,  and  that  we  wish  to  increase  its 
temperature  by  dT  degrees  centigrade  and  still  keep  it 
saturated.  We  can  proceed  in  the  following  way: 

We  communicate  to  it  a  quantity  of  heat  dQl  which  is  just 
sufficient  to  raise  its  temperature  by  dT  degrees.  The  vapor 
is  then  superheated,  that  is  to  say,  its  density  is  smaller  than 
the  density  corresponding  to  its  saturated  state  at  that 
temperature  T -}- dT.  In  order  to  increase  the  density  so 
as  to  make  it  equal  to  the  density  of  saturation  at  the 
temperature  T -\- dT,  we  now  compress  it  isothermally 
until  the  proper  density  is  reached.  Let  the  heat  developed 
by  this  compression  be  dQ^y  then  dQl  —  dQ9  =  tlie  heat 
which  must  be  supplied  to  1  kg.  of  the  saturated  vapor  in 


REVERSIBLE  CYCLES  IN  OASES  AND    VAPORS.       61 

order  to  raise  its  temperature  by  dT  degrees  and  still  keep 
the  vapor  saturated  at  this  increased  temperature. 

*l  1T     -  is  denoted  by  H  and  called  the  specific  heat  (in 

kg.-m.)  of  saturated  vapor.  It  is  evidently  equal  to  the  heat 
which  would  have  to  be  supplied  to  one  kilogramme  of  satu- 
rated vapor  in  order  to  raise  its  temperature  by  one  degree  and 
maintain  it  in  saturated  condition,  if  the  ratio  of  the  supply 
of  heat  to  the  rise  of  temperature  remained  constant. 

Hence  in  our  problem  above  the  part  of  dQ  which  is  ex- 
pended in  raising  the  temperature  of  the  m  kg.  of  saturated 
vapor  without  superheating  and  without  increasing  the  quan- 
tity of  m  will  be  mHdT. 

The  total  heat  contained  in  the  mass  will  be  increased  be- 
cause the  temperature  of  the  mass  is  increased.  The  rate  at 
which  the  heat  of  the  mass  increases  with  the  increase  of 
temperature,  everything  else  remaining  the  same,  is  expressed 

in  the  well-known  notation  —^      Hence  the  increment  in  the 

o  I 

heat  due  to  the  increment  dT  of  the  temperature  is  ^dT. 

This  is  evidently  equal  to  the  heats  mentioned  under  1  and  2. 
We  can  therefore  write 


-dr  =(M-m)  CdT  +  mHdT, 

O-L 


or 


62  THERMODYNAMICS  OF 

DEFINITIONAL  PROCESS  (B). 

Let  the  piston  rise  so  as  to  allow  a  small  quantity 
dm  of  the  liquid  to  evaporate.  In  consequence  of  this 
evaporation  the  temperature  of  the  mass  begins  to  sink  ; 
we  therefore  supply  heat  in  order  to  maintain  it  con- 
stant. Let  dQ  be  equal  to  the  heat  supplied  to  the  total 
mass  during  the  isothermal  evaporation  of  the  quantity 

dm.     The  ratio  -^  is  denoted  by  p  and  called  the  heat  of 
dm 

evaporation.  It  evidently  means  the  number  of  units  of  heat 
which  must  be  supplied,  in  order  to  evaporate  isothermally 
one  kg.  of  water  at  T  deg.  centig.,  and  the  pressure  of  satu- 
rated vapor  at  that  temperature  into  saturated  vapor.  Let 

—  denote  the  rate  at  which  the  heat  of  the  whole  mass  in- 
o»i 

creases   when   the  water    is    evaporated   isothermally,   then 

^dm  is  evidently  the  total  increment  of  heat  when  dm  unit 
dm 

weights  of  the  liquid  pass  isothermally  into  that  many  units 
of  saturated  vapor.  But  we  have  just  seen  that  this  is  also 
equal  to  pdm.  Hence 


--^dm  =  pdm, 
cm 


or 


REVERSIBLE  CYCLES  IN  OASES  AND    VAPORS.       63 

The  two  equalities  obtained  by  the  definitional  processes 
(A)  and  (B)  enable  us  now  to  deduce  a  very  elegant  state- 
ment of  the  two  laws  of  thermodynamics  for  saturated  vapors. 


FORMS  OF  STATEMENT  OF  THE  Two   LAWS   OF   THERMO- 
DYNAMICS FOR  SATURATED  VAPORS. 

We  start  with  the  general  statements  of  these  two  laws: 


and 

dQ  =  TdS  .  . 

By  carrying  out  the  indicated  differentiation,  remember- 
ing that  T  and  m  are  our  variables  —  that  is  to  say,  remember- 
ing that  Q,  U,  S,  and  v  are  in  our  discussion  of  saturated 
vapors  functions  of  T  and  m  and  of  nothing  else  —  we  obtain 
the  following  relations  : 

We  commence  with  the  first  law. 


This  is  an  identical  equation  in  dm  and  dT\  hence  we  must 
have,  according  to  well-known  rules  in  algebra, 


64  THERMODYNAMICS  OF 

Comparing  these  two  relations  with  the  expressions  for 
^  and  --  obtained  by  the  definitional  processes  (A)  and  (B), 
we  infer  that 


30 


It  must  be  observed  that  these  two  equalities  are  nothing 
more  nor  less  than  a  slightly  different  mathematical  statement 
of  the  first  law.  Nor  will  any  other  mathematical  relation 
which  we  can  derive  from  these  two  equalities  by  any  mathe- 
matical operations  whatever  contain  a  single  grain  of  truth 
which  is  not  already  contained  in  the  original  statement  of 
that  law.  What  do  we  gain,  then,  by  going  through  all 
these  tedious  mathematical  operations  ?  Do  we  arrive  at  a 
simpler  statement  of  this  law  ?  No,  we  do  not.  Mathemati- 
cally speaking,  the  expression 


is  the  simplest  form  in  which  that  law  can  be  stated.  Before 
answering  the  above  qrestion  let  us  first  perform  the  last 
mathematical  operation  and  obtain  the  final  form  of  state- 
ment of  the  first  law.  Differentiate  the  first  of  the  above 
equations  with  respect  to  m  and  the  second  with  respect  to  T, 
and  subtract  the  first  from  the  second  so  as  to  eliminate  U. 


HEVEKSIBLE  CYCLES  IN  GASES  AND    VAPORS.       65 

Remember,  however,  that  p,  H,  M,  and  C  are  independent 
of  m.     We  obtain 


Eemembering  now  that  v  =  um  -J-  Ma,  hence  — -  =  u9  we 
obtain  finally 

/  TT n\  ij.    P  (J\ 


This  is  the  statement  of  the  first  law  of  thermodynamics 
for  saturated  vapors  at  which  ive  were  aiming  continually 
during  our  tedious  mathematical  transformations.  Now 
in  what  respect  is  this  statement  superior  to  the  general 
statement  dQ  =  d  U -{- pdv  ?  Evidently  in  this:  it  con- 
tains nothing  but  physical  constants  of  the  liquid  and  its 
saturated  vapor,  most  of  which,  as  we  shall  presently  see,  are 
capable  of  exact  experimental  measurement.  A  very  famous 
scientist  said  once  that  a  department  of  human  knowledge 
becomes  an  exact  science  when  it  can  express  its  laws  in 
terms  of  things  which  are  capable  of  exact  experimental 
measurement.  The  intrinsic  energy  U,  to  be  sure,  is  a  thing 
which  we  can  and  did  express  in  terms  of  a  definite  unit,  the 
kilogramme-meter;  we  can  also  give  a  perfectly  satisfactory 
definition  for  it.  We  can  evidently  define  it  in  two  ways, 


66  THERMODYNAMICS  OF 

which  we  may  call  the  absolute  and  the  relative  method  of 
definition.  The  absolute  definition  describes  U  as  the  energy 
which  we  can  obtain  from  the  body  by  a  series  of  processes 
which  would  reduce  the  body  to  a  state  in  which  its  intrinsic 
energy  is  zero.  The  relative  definition  describes  U  as  the 
energy  which  we  can  obtain  from  a  body  by  a  series  of  pro- 
cesses which  will  reduce  the  body  to  a  state  which  we  call 
the  normal  state;  and  we  can  select  any  state  for  our  normal 
state.  For  instance,  we  can  define  the  intrinsic  energy  of  a 
kilogramme  of  saturated  water-vapor  at  any  temperature  as 
the  energy  which  we  can  obtain  from  it  by  reducing  it  to  a 
kilogramme  of  water  at  0°  C.  and  a  pressure  equal  to  the 
tension  of  its  saturated  vapor  at  that  temperature.  Still, 
when  it  comes  to  exact  experimental  measurement  it  is  not 
U  that  we  measure  but  the  physical  constants  p,  H,  C,  and 
•it,  and  from  the  measurements  of  these  the  relative  value  of 
U  can  then  in  a  very  limited  number  of  cases  be  obtained  by 
calculation.  Hence  the  desirability  of  stating  the  first  law  of 
thermodynamics  of  saturated  vapors  in  terms  of  these  physi- 
cal constants  which  are  capable  of  exact  experimental  meas- 
urement instead  of  in  terms  of  U9  a  quantity  which  is  not 
capable  of  direct  experimental  measurement.  It  is  clear 
now  that  the  ultimate  object  of  all  our  long  mathematical 
operations  was  to  rid  ourselves  by  a  process  of  elimination 
of  the  quantity  U. 

We  proceed  now  to  perform  the  same  process  of  elimina- 
tion on  the  general  statement  of  the  second  law.  Carrying 
out  the  differentiations  indicated  in 

i 

d    =  TdS, 


REVERSIBLE  CYCLES  IN  OASES  AND   VAPORS.       67 

and  remembering  that  T  and  m  are  our  independent  variables, 
we  obtain 


This  is  an  identical  equation  in  ^7*  and  dm,  and  therefore 
the  following  relations  will  hold  true : 


Comparing  these  two  relations  with  those  obtained  by  the 
definitional  processes  (A)  and  (B)  we  obtain 


Differentiating  now  the  first  equation  with  respect  to  m 
and  the  second  with  respect  to  T  and  then  subtracting  the 
first  from  the  second  in  order  to  eliminate  8,  we  obtain 


dp 


68  THERMODYNAMICS  OF 

But  since 

9wi  ~~  9wi  ~~  **' 
we  obtain 


(II) 


This  is  our  final  statement  of  the  second  law  of  thermo- 
dynamics for  reversible  processes  in  saturated  vapors.  It 
contains,  just  like  the  statement  (I)  of  the  first  law,  a  mathe- 
matical relation  between  the  experimentally  measurable 
physical  constants  of  the  liquid  and  its  saturated  vapor. 

By  combining  (I)  and  (II)  we  obtain  another  very  impor- 
tant relation  between  these  constants,  viz., 


(Ill) 


Observation.  —  It  is  well  to  call  your  attention  here  to  a 
remark  which  was  made  in  connection  with  our  deduction  of 
the  differential  statement  of  the  second  law, 

dQ  =  TdS, 
from  the  integral  statement 


REVERSIBLE  CYCLES  IN  OASES  AND   VAPORS.       69 

The  remark  was  to  the  effect  that  the  integral  statement 
described  the  total  effect  of  a  series  of  processes  which  taken 
together  constitute  a  reversible  cycle,  without  telling  us 
anything  about  the  separate  effects  of  each  particular  process 
of  that  series.  The  differential  statement,  on  the  other 
hand,  describes  completely  every  minute  step  in  the  progress 
of  the  reversible  cycle.  The  relations  (II)  and  (III)  illus- 
trate the  truth  of  this  remark  very  forcibly;  for  they  tell  us 
that  during  every  interval  of  time,  no  matter  how  small,  the 
reversible  operations  will  progress  in  such  a  way  as  to  main- 
tain those  relations  between  the  physical  constants  and  the 
variable  co-ordinates  of  the  liquid  and  its  vapor  which  are 
given  by  (II)  and  (III).  But  do  not  forget  that  relation  (II) 
and  to  same  extent  relation  (III)  also  is  only  another  way  of 
stating  the  differential  form  of  the  second  law, 

dQ  =  TdS. 

More  than  this.  The  integral  statement  limits  us  to  the 
study  of  those  cycles  which  are  composed  of  reversible  opera- 
tions only.  The  differential  statement  enables  us  to  break 
through  these  limits  and  study  cycles  consisting  of  operations 
some  of  which  are  not  reversible.  For  it  is  evident  that  we 
can  apply  (II)  and  (III)  to  every  reversible  operation  of  the 
cycle,  and  the  irreversible  operations  we  can  attack  by 
the  first  law  or  in  any  other  way  that  we  may  consider  cor- 
rect and  convenient.  This  has  a  very  important  technical 
significance,  for  we  shall  presently  see  that  the  cyclic  opera- 
tions in  those  types  of  steam  and  other  caloric  engines  which 
prevail  to-day  are  by  no  means  reversible  in  all  their  parts. 


70  THERMODYNAMICS  OF 

Since  numerical  values  of  the  physical  constants  p,  H,  and 
C  have  been  tabulated  by  various  experimentalists  in  terms 
of  the  kg.  calorie  and  not  the  kg.  meter  as  the  unit,  it  is 
advisable  to  substitute  in  our  equations  (I),  (II),  and  (III) 
the  numerical  values  of  p,  H,  and  (?in  terms  of  the  kg.  caloric 
as  unit.  Let 

r  =  the  numerical  value  of  p  when  the  unit  is  a  kg.  calorie ; 
£  __    (t          ((  ((      ((  n       ((       ((      ((     ( 


then 


P  c  H      T, 

J  =  r>    J=c>   T  =  h> 


hence  we  obtain  from  (I),  (II),  and  (III)  by  these  substitu 
tions 

__  +  c_;i  =  ___,      ....    (Ia 


> 


Time  three  equations  are  our  fundamental  equations  in  the 
thermodynamics  of  reversible  processes  in  saturated  vapors. 
We  now  proceed  to  the  next  part  of  our  programme,  and 


REVERSIBLE  CYCLES  IN  OASES  AND   VAPORS.       71 

that  is  the  numerical  calculation  of  the  physical  constants  of 
saturated  vapors. 

NUMERAL  CALCULATION  OF  THE  PHYSICAL  CONSTANTS  OF 
SATURATFED  VAPORS. 

Our  three  fundamental  equations  (Ia),  (IIa),  and  (IIP)  repre- 
sent two  independent  relations  [since  (IIP)  has  been  deduced 
from  (Ia),  that  is  the  first  law,  and  (IIa),  that  is  the  second 
law  of  thermodynamics]  between  five  unknown  quantities, 

viz.,  r,  h,  u,  c,  and  —^-.     Hence  they  enable  us  to  calculate 

any  two  of  these  quantities  when  the  remaining  three  are 
known.  That  is  to  say,  three  of  these  physical  constants  of 
saturated  vapors  must  be  determined  experimentally  and  the 
other  two  can  then  be  calculated  by  means  of  the  fundamental 
laws  of  thermodynamics.  What  is  meant  by  experimental 
determination  is  simply  this:  It  must  be  found  by  experi- 
mental investigation  in  what  way  these  physical  constants 
depend  on  the  variable  co-ordinates  of  the  body.  In  our  dis- 
cussion of  the  general  physical  properties  of  saturated  vapors 
we  saw  that  all  the  physical  constants  of  such  a  vapor  depend 
on  the  temperature  only,  hence  the  object  of  an  experimental 
investigation  will  be  to  express  these  physical  constants  as 
functions  of  the  temperature  T.  Three  of  the  above  con- 
stants having  been  experimentally  determined,  the  remaining 
two  can  then  be  also  expressed  as  functions  of  the  temperature 
by  simple  calculation,  starting  with  our  fundamental  equa- 
tions (Ia),  (IIa),  and  (IIP). 

In  most  cases  r,  c,  and  p  have  been  determined  experiment- 


72  THERMODYNAMICS  OF 

ally,  so  that  it  is  generally  //  and  u,  that  is,  the  specific  heat 
and  the  specific  volume  of  saturated  vapors,  that  we  have  to 
calculate. 

We  now  proceed  to  carry  out  these  calculations  for  a  par- 
ticular vapor,  and  select  water  vapor  for  reasons  given  above. 

Regnault  in  his  classical  researches  (Relation  des  experiences, 
Mem.  de  1'Acad.  t.  xxi,  1847,  etc.)  made  the  quantities  c,  r, 
and  JP,  that  is,  the  specific  heat  of  water,  the  heat  of  evapora- 
tion, and  the  tension  of  saturated  vapor  of  water,  the  subject 
of  very  careful  experimental  investigations.  His  results  have 
been  verified  by  all  investigators  in  that  field  of  physical  re- 
search. We  shall,  therefore,  use  the  values  obtained  from 
the  results  of  his  investigations. 

The  specific  heat  of  water  is  expressed,  according  to 
Regnault's  experimental  data,  by  the  following  formula : 

c  =  1  +  .00004^  +  .0000009*", 

- 

where  t  is  the  temperature  in  degrees  centigrade  above  the 
freezing  point. 

For  the  heat  of  evaporation  experimental  data  furnish  the 
following  relation : 


r  =  606.5  +  .305^  -  C  cdt. 

t/O 


Substituting  the  value  of  c  and  carrying  out  the  integration, 
we  obtain 

r  =  606.5  -  .695^  -  .00002?  -  .0000003?. 


REVERSIBLE  CYCLES  IN  OASES  AND    VAPORS.       73 


For  this  long  formula,  Clansius  (Mechanische  Warmetheorie, 
vol.  i.  p.  137)  substitutes  the  following: 

r=  607 -.708*. 

To  show  the  agreement  between  the  values  of  r  obtained 
from  these  two  formulae  the  following  table  is  given  for  com- 
parison : 


t 

0° 

50° 

100° 

150° 

200° 

T  according  to  Regnault's  form  .... 

606  5 

571  6 

586  5 

500  7 

464  3 

T         "          "  Clciusius'  form 

607 

571  6 

536  2 

500  8 

465  4 

The  agreement  is  excellent.      We  shall,  therefore,  employ 
Clausius*  simpler  expression. 

Taking  the  experimental  data  of  Regnault's  researches, 
Clausius*  constructed  the  following  table  for  the  tension^? 
of  saturated  water- vapor  at  various  temperatures: 


t  in  C.  °  by 
Air  Therm. 

p  in  mm. 
of  Hg. 

t  in  C.  °  by 
Air  Therm. 

p  in  mm. 
ofHg. 

-20 

.91 

110 

1073.7 

-10 

2.08 

120 

1489 

0 

4.60 

130 

2029 

10 

9.16 

140 

2713 

20 

17.89 

150 

3572 

30 

31.55 

160 

4647 

40 

54.91 

170 

5960 

50 

91.98 

180 

7545 

60 

148.79 

190 

9428 

70 

283.09 

200 

11660 

80 

854.64 

210 

14308 

90 

52545 

220 

17390 

100 

760 

230 

20915 

*  Mechanische  Warmetheorie,  vol.  i.  p.  149. 


74  THERMODYNAMICS  OF 

OBSERVATION  1.  To  transform  p  from  pressure  in  mm. 
Hg  to  pressure  in  kg.  per  sq.  m.  we  simply  remember  that 
760  mm.  barometric  pressure  is  equivalent  to  10,333  kg.  per 
square  meter.  If  we  now  wish  to  know  how  many  kg.  per 
sq.  m.  correspond  to,  say,  11,660  mm.  barometric  pressure,  we 
simply  put  down  the  proportion 

p:  10,333  ::  11,660:760, 

where  p  is  the  pressure  in  kg.  per  sq.  m.,  corresponding  to 
11,660  mm.  barometric  pressure. 

10333  X  1166 
p  =  -      —  —  —          -  Io8,530  kg.  per  sq.  m. 

(about  229  pounds  per  square  inch). 

This,  then,  is  the  pressure  of  saturated  water-vapor  at  200°  C. 
OBSERVATION  2.  If  we  refer  the  tension  p  and  the  tem- 
perature t  of  the  above  table  to  a  set  of  rectangular  axes, 
measuring  off  t  as  the  abscissae  and  the  corresponding  p's  as 
the  ordinates,  we  obtain  the  curve  as  in  diagram  on  opposite 
page. 

The  value  of  the  differential  coefficient  —  (which  is  evi- 

dt    \ 

dently  the  same  as   ~j    at  any  temperature  represents  the 


tangent  of  the  angle  which  the  tangent  line  to  the  curve  at 
the  point  corresponding  to  that  temperature  makes  with  the 
axis  of  t.  An  inspection  of  the  curve  shows  that  this  angle 
remains  practically  constant  for  considerable  intervals  of 
temperature,  certainly  for  intervals  of  10°, 


IRBVERSTELB  CYCLES  IN  GASES  AND    VAPORS.       75 


I  dp       d 
' 


a 


0  15  30  45  60  75  9  10        ' 

Consider  now  any  interval  of  10°  and  let 

-JT  =  OL  for  that  interval. 
at 


. 

=  ~  for  tlie  sam^  interval 


76  THERMODYNAMICS  OF 

This  relation  enables  us  to  calculate  in  a  very  simple  manner 
the  quantity  -  ~  for  any  temperature.  This  is  a  quantity 
which  we  shall  need  presently.  Suppose  we  wish  to  calculate 

(—  ~]  ,  that  is,  the  value  of  -  -j-  for  the  temperature  of 
\p  dt  /25  p  dt 

t  =  25°.  We  select,  therefore,  that  10°  interval  the  mean 
temperature  of  which  is  25°,  that  is,  the  interval  between  20° 
and  30°.  We  obtain 


log />.. -log  A.  =  «   /    dl. 

P 


The  mean  value  of  p  within  the  limits  of  integration  is  very 
nearly  the  value  of  p  at  25°,  that  is,  pw 


i  <* 

•'•  log  A.  -  log  p,,  =  ~-   I  dt  =  ---. 


But  «=m  ; 


1  fdp\        logffso-logAo  =  Log  ^30- Log  ao 
10  J/10  ' 


where  Log  stands  for  Briggs'  logarithm,  and  Mis  the  modulus 
of  that  system. 

We  are  ready  now  to  take  up  the  calculation  of  the  re- 
maining physical  constants,  that  is,  to  find  the  values  of  the 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.       77 

specific  heat  and  the  specific  volume  of  saturated  water-vapor 
for  various  temperatures  within  those  limits  within  which  the 
other  physical  constants  were  studied  experimentally. 


1.  Calculation  of  h. 
From  the  second  law 

^r    i  TJ  -    r 

we  obtain 

dr  .  r 


Kegnault's  experiments  gave  us 


r  =  606.5  +  .305^-    /  cdt 


.-.   ~  =  . 
at 


Therefore 


78 


THERMODYNAMICS  OF 


In  the  last  term  of  the  expression  for  li  we  shall  substitute 
for  r  the  simple  expression  given  by  Clausius  and  obtain 


Ji  =  .305  - 


607  -.708; 
273     T~* 


The  following  table  contains  the  calculated  values  of  li  for 
several  temperatures: 


t 

0° 

50° 

100° 

150° 

200° 

h 

-  1.916 

-  1.465 

-  1.133 

-.879 

-.676 

The  physical  meaning  of  this  negative  specific  heat  will  be 
considered  further  on  in  connection  with  adiabatic  expansion 
of  saturated  water-vapor. 

We  pass  on  to  the  calculation  of  the  specific  volume  of 
saturated  water-vapor. 


2.  Calculation  of  the  Specific  Volume  of  Saturated 
Water- vapor. 

The  first  question  to  consider  is,  whether  in  its  saturated 
state  water- vapor  follows  the  law  of  Mariotte-Gay-Lussac, 
and  if  it  does  not,  to  what  extent  and  in  what  way  deviations 
occur. 

Making  use  of  the  fundamental  relation  (IIIa),  we  find, 
after  replacing  (s  —  a)  for  u, 


_  T(s  —  a)  dp 
T~     ~dt 


REVERSIBLE  CYCLES  IN  GASES  AND   VAPORS.       79 
or 

273r 


Mp  -   </(273  +  0' 


If  the  Mariotte-Gay-Lussac  law  were  applicable  to  satu- 

/n  Q 

rated  water  -  vapor,  then   from  ~  =  const,  we  should  have 


ps  X  273    _ 

Since  s  differs  but  little  from  (s  —  cr),  we  should  have 

=  const. 


Comparing  this  to  the  equation  just  obtained  from  the  funda- 
mental relation  (IIP),  we  see  that  if  the  Mariotte-Gay-Lussac 
law  were  applicable  to  saturated  water-vapor  we  should  have 

273r      _  p(s  -  o-)273  _ 
~ldp-  ": 

p  dt 


Having  calculated  -  -^-  by  the  method  described  above,  and 
r  from  the   formula  r  =  607  —  .708^,  and  substituted   their 


80 


THERMODYNAMICS  OF 


values  in  the  last  equation,  Clausius  obtained  the  following 
table: 


_*.  CVC/ 

'  J*"         "'273 

-M* 

tin  C.°by  Air 
Thermometer. 

Calculated  from 
Regnault's  data. 

Calculated  from 
Clausius'  formula. 

5 

30.93 

30.46 

15 

30.60 

30.38 

25 

30.40 

30.30 

35 

30.23 

30.20 

45 

30.10 

30.10 

55 

29.98 

30.00 

65 

29.88 

29.88 

75 

29.76 

29.76 

85 

29.65 

29.63 

95 

29.49 

29.48 

105 

29.47 

29.33 

115 

29.16 

29.17 

125 

28.89 

28.99 

135 

28.88 

28.80 

145 

28.65 

28.60 

155 

28.16 

28.38 

165 

28.02 

28.14 

.    175 

27.84 

27.89 

185 

27.76 

27.62 

195 

27.45 

27.33 

205 

26.89 

27.02 

215 

26.56 

26.68 

225 

26.64 

26.32 

REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.       81 

The  figures  given  in  the  column  headed  ' '  Calculated  from 
Clausius'  formula  "  will  be  explained  presently. 

We  can  easily  see  from  this  table  that  the  Mariotte-Gay- 

1  273 

Lussac  law  does  not  hold  true;  for  -jp(s  —  0-)-— -.  dimin- 

J  £  to  -j-  t 

ishes  steadily  with  the  increase  of  temperatures;  evidently 
because  the  specific  volume  of  saturated  steam  diminishes 
more  rapidly  than  the  pressure  increases  with  increase  of 
temperature.  Clausius  found  that  this  formula  could  be 
reduced  to  the  more  simple  one,  viz., 


=  m  —  neat, 

where        m  =  31.549, 
n=    1.0486, 
a  =      .007138. 

By  means  of  this  formula  Clausius  calculated  the  figures 
given  in  the  column  headed  "  Calculated  from  Clausius'  for- 
mula." 

After  this  brief  digression  we  return  to  the  calculation  of 
the 

Specific  Volume  of  Saturated  Steam. 

We  had 

1  273 


82  THERMODYNAMICS  OF 

Since  we  commit  an  error  of  a  small  fraction  only  of  one  per 
cent  by  neglecting  cr  in  comparison  with  s,  we  may  write 

_  J  273  +  t 

-  p(«*      ™*  )    m  , 


Let  v'  =  volume  of  a  kilogr.  of  air  at  a  pressure  p  and 
temperature  t,  then  will 


For  a  given  temperature  t,  say  the  freezing  temperature, 
this  ratio  can  be  determined  experimentally,  as  will  be  ex- 
plained presently. 

Denoting  the  ratio  —  ,  at  the  freezing  temperature  by  [—  ,) 
v  \v  /„, 

we  shall  have 


s  _  /  s  \  m  —  neat 
v'       w  /0  m  —  n 

The  value  of  f  —  ,  j   can  be  obtained  easily  by  the  following 


method  : 


REVERSIBLE  CYCLES  IN  GASES  AND   VAPORS.       83 

Experiment  tells  us  that  saturated  water-  vapor  at  low  tem- 
peratures obeys  the  Mariotte-Gay-Lussac  law  so  nearly  that 
we  may  use  the  law  in  practical  calculations  without  commit- 
ting an  appreciable  error.  On  that  hypothesis  the  density  of 
steam  at  0°  C.  is  easily  obtained. 

2  cub.  m.  H  at  0°  0.  at  a  given  pressure  weigh  2  X  .06926 
as  much  as  1  cub.  m.  of  air  at  the  same  temperature  and 
pressure; 

1  cub.  m.  0  at  0°  0.  at  a  given  pressure  weighs  1  X  1.1056 
as  much  as  1  cub.  m.  of  air  at  the  same  temperature  and 
pressure; 

At  low  temperature,  2  cub.  m.  H  and  1  cub.  m.  0  unite  into 
2  cub.  m.  saturated  water-  vapor  (very  nearly). 

.*.  2  cub.  m.  H20  at  0°  C.  at  a  given  pressure  weigh 
2  X  .06926  -f  1.1056  as  much  as  1  cub.  m.  of  air  at  the 
same  temperature  and  pressure; 

1    cub.    m.    of   H20   at   0°   C.   at   a  given   pressure   weighs 

2  X  .06926  +  1.1056 

—  —  -  as  much  as  1  cub.  m.  of  air  at  the 

same  temperature  and  pressure. 


__ 

*  V/0      |(2  X  .06926  +  1.1056)       .622' 
Therefore 

s          1     m  —  neat 

v'  ~  .1J23   m  -  n  ' 


84  THERMODYNAMICS  OF 

which  can  also  be  expressed  by  a  more  convenient  formula 
given  by  Clausius,  thus: 


In  this  formula 

M=  1.663, 

N=    .05527, 
ft  =  1.007164. 

This  formula  was  used  by  Clausius  to  calculate  the  specific 
volume  of  saturated  water-vapor  at  various  temperatures. 
The  results  are  given  in  the  table  on  page  85.  Parallel  with 
these  figures  are  given  the  figures  obtained  experimentally  by 
two  English  engineers,  Fairbairn  and  Tate.* 

It  must  be  observed  that  for  every  temperature  we  must 
'mow  first  the  pressure  of  saturated  vapor  at  that  tempera- 
ture, and,  inserting  that  pressure  in  pv'  =  RT,  calculate  v'. 

Observe  the  close  agreement  between  experiment  and 
theory  as  worked  out  by  Clansius;  observe  also  the  enormous 
increase  of  the  density  of  steam,  with  rising  temperature. 

There  is  a  temperature  at  which  the  density  of  steam  is 
the  same  as  that  of  water  at  the  same  temperature.  This  is 
called  the  critical  temperature  of  water.  Above  the  critical 
temperature  the  vapor  cannot  be  liquefied  by  pressure  alone. 

It  is  highly  probable  that  all  bodies  have  a  critical  tem- 
perature. In  the  case  of  the  so-called  perfect  gases  the  criti- 

*  Transactions  of  the  Royal  Society  of  London,  I860,  vol.  150,  p.  185. 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.       85 


Table  of  Specific  Volume  of  Saturated  Water-vapors. 


t  in  C.° 

From 
Clausius-'  Form. 

Fuirbaini  and  Tate's 
Experimental  Table. 

58.21 

8.23 

8.27 

68.52 

5.29 

5.33 

70.76 

4.83 

4.91 

77.18 

3.74 

3.72 

77.49 

3.69 

3.71 

79.40 

3.43 

3.43 

83.50 

2.94 

3.05 

86.83 

2.60 

2.62 

92.66 

2.11 

2.15 

117.17 

0.947 

0.941 

118.23 

0.917 

0.906 

118.46 

0.911 

0.891 

124.17 

0.769 

0.758 

128.41 

0.681 

0.648 

130.67 

0.639 

0.634 

131.78 

0.619 

0.604 

134.87 

0.569 

0.583 

137.46 

0.530 

0.514 

139.21 

0.505 

0.496 

141.81 

0.472 

0.457 

142.36 

0.465 

0.448 

144.74 

0.437 

0.432 

cal  temperature  is  very  low;  hence  the  necessity  of  cooling  in 
the  liquefaction  of  gases.* 

We  now  come  to  the  last  and  most  important  part  of  our 
discussion  of  the  application  of  the  two  laws  of  thermo- 
dynamics to  the  study  of  reversible  processes  in  saturated 
vapors  in  general  and  water-vapor  in  particular,  that  is,  to 
the  discussion  of  adiabatic  and  isothermal  expansion  of  satu- 
rated vapors. 


*  See  tiie  researches  of  Aveuarius  in  Poggeudorf'g 

^x'bt 


86  THERMODYNAMICS  OF 


ADIABATIC  EXPANSION  OF  SATURATED  WATER-VAPOR. 

We  commence  with  the  deduction  from  our  fundamental 
relations  of  a  mathematical  expression  which  simplifies  con- 
siderably our  discussion  of  adiabatic  expansion. 

By  the  definitional  processes  (A)  and  (B)  we  obtained 


rffli  =  \m(H-  C)  +  MC}dT+  pdm. 

o  1  cm 

From    the    second    law    (Fundamental    Kelation   II)    we 
obtain 


Substituting  this  value  of  (H  —  C)  in  the  last  equation,  we 
derive  the  relation 

dQ  =  {  «(|y-  f)  +  MC^  dT+  pdm. 


Considering  now  that  p,  the  heat  of  evaporation,  depends 
on  temperature  only,  it  is  evident  that 


REVERSIBLE  CYCLES  IN  OASES  AND    VAPORS.       87 


••        7ri  —  pdm  =  d(mp)-y 

O-L 


...  dQ  =  d(mp)  +  (MO  -  "f)dT, 
or 

dQ  =  Td(^}  +  MCdT. 


This  is  the  expression  which  we  started  out  to  deduce. 
In  adiabatic  expansion  dQ  =  0,  hence 


or 


mr      m1r1  _         M  C  cdT 
•'•~T~~T~        ~MJ    ~T' 

1  *J  T* 


Calculation  is  simplified  very  much  and  the  error  com- 
mitted is  very  small  if  we  put  c  =  constant,  during  the  expan- 
sion, provided  that  the  expansion  does  not  extend  over  a  very 
large  interval  of  temperature.  In  practice  this  never  occurs. 

mr      mr  T 

~~~^=+          g     ' 


or 


T 

m  =  - 


THERMODYNAMICS  OF 

(a)  Condensation  during  Adialatic  Expansion  of  Saturated 
Water-vapor. 

A  very  interesting  but,  from  an  economical  standpoint, 
a  very  objectionable  phenomenon  takes  place  when  satu- 
rated water-vapor  expands  adiabatically.  Owing  to  the  fact 
that  during  such  an  expansion  the  temperature  (in  con- 
sequence of  external  work  done  by  the  expansion)  sinks  more 
rapidly  than  the  density  diminishes,  the  -vapor  condenses. 
Hence  in  order  to  prevent  this  condensation  it  would  be 
necessary  to  supply  heat  during  the  expansion.  This,  then, 
is  the  physical  meaning  of  the  negative  specific  heat,  as  found 
above,  of  saturated  water-vapor.  During  adiabatio  compres- 
sion the  vapor  would,  of  course,  become  superheated.  The 
last  equation  enables  us  to  calculate  the  amount  of  this  con- 
densation. For  example,  suppose  that  m1  kilogrammes  of 
saturated  water-vapor  at  initial  temperature  T^  are  enclosed 
in  a  cylinder  by  a  piston  which  can  glide  up  and  down  with- 
out friction.  Let  there  be  no  liquid  mass  in  the  cylinder. 
Hence  M=  7/2,,  and  as  the  piston  goes  up  slowly  owing  to  the 
gradual  diminution  of  the  pressure,  we  shall  have  at  any  mo- 
ment 

m       T 


Starting  with  initial  temperature  Tl  =  150°  +  273,  Clausius* 
calculated  the  ratio  of  the  vapor  weight  m  at  various  tem- 
peratures T  to  the  initial  weight  mr  The  following  table 
contains  the  interesting  results  of  this  calculation: 

*  Mechanische  Warmeiheorie,  vol.  i.  p.  164. 


REVERSIBLE   CYCLES  IN  OASES  AND    VAPORS.       89 


i 

150° 

,*. 

100° 

75° 

50° 

25° 

m 

1 

.956 

.911 

.866 

.821 

.776 

The  amount  of  condensation  is,  therefore,  very  consider- 
able. 

This  phenomenon  contributes  to  what  is  known  in  steam- 
engineering  under  the  name  of  cylinder  condensation,  a  pro- 
cess which,  as  is  easily  seen,  pulls  down  the  output  of  the 
steam-engine.  Other  and  probably  quite  as  serious  causes 
contribute  to  this  objectionable  process  in  the  cylinder  of  a 
steam-engine.  Various  devices  have  been  suggested  and 
tried  in  order  to  overcome  this  evil,  but  the  discussion  of 
these  is  outside  of  the  limits  of  this  course. 

(/?)   Calculation  of  the  External  Work  done  during  an 
Adiabatic  Expansion. 


W 


=    /   pdv. 

tx  Vj 

But  v  =  mu  +  Jfcr. 

/.  r>^v  =pd(mu)  =  d(mup)  —  mu-^d'j 

O-L 

From  our  fundamental  equation  (III)  we  have 


. 


mp 


.:pdv=d(mup)--£-dT. 


90  THERMODYNAMICS   OF 

Since  v  changes  adiabatically,  we  can  make  use  of  the  re- 
lation deduced  in  the  last  paragraph  when  we  considered  the 
process  of  condensation  during  adiabatic  expansion.  The  re- 
lation is 


or 


^-dT  -  d(mp)  =  MCdT. 
.\pdv  =  d(mup)  —  d(mp)  —  MCdT. 

.V  W  =  /    d(mup)  -  /    d(mp)  -  MC        dT 
JVl  JVl  JT, 

=  mup  -  m1u1p1  -  [mp  -  w^pj  -f  MC(Tl  -  T). 

Since  the  quantities  p  and  C  are  given,  by  the  experimental 
formulae  discussed  above,  in  kg.-calories,  it  is  preferable  to  ex- 
press W thus: 

W=  mup  -  vnlulpl  —  J{(mr  —  m^)  —  Mc(T^  —  T)\. 

This  formula  enables  us  to  calculate  the  external  work  W 
in  kg.-meters  when  a  certain  initial  quantity  ml  of  saturated 
water-vapor  at  temperature  T1 ,  being  in  contact  with  a  quan- 
tity M  —  m^  of  its  own  liquid  at  the  same  temperature,  ex- 
pands adiabatically  until  its  temperature  has  gone  down  to  a 
temperature  T.  Thus: 


REVERSIBLE  CYCLES  IN  OASES  AND   VAPORS.      91 

Since  Tl  and  T  are  given,  pl  and  p  can  be  found  by  a  refer- 
ence to  Kegnault's  table  of  vapor  tensions.  Since  m1  and  M 
are  also  given,  JMc(Tl  —  T)  and  Jmj\  can  be  easily  calcu- 
lated. There  still  remain  mr  and  mu  to  be  calculated.  The 
quantity  mr  we  have  already  calculated  in  the  preceding 
paragraph  and  found 


mr  = 


The  quantity  mu  can  be  easily  found  thus : 
From  the  fundamental  relation  (IIP)  we  have 

T  dp 


Substituting  this  value  of  mr  in  the  last  equation,  we  ob- 
tain 

J  (mlrl  '. 

W 

The  quantity  -—,  is  the  tangent  of  the  angle  which  the 

tangent  line  to  the  curve  of  vapor  tension,  given  above,  at 
the  point  corresponding  to  temperature  T  =  273  -f-  t,  makes 
with  the  axis  of  t,  and  can  be  easily  found  either  graphically 
or  by  calculation,  thus: 

Suppose   we  wish    to  calculate  - —,  for.  the  temperature 


92  THERMODYNAMICS  OF 

t  =  110°.     On  account  of  the  gradual  ascent  of  the  curve  of 
vapor  tension,  we  have 


«.~         10 

Referring  now  to  Regnault's  table  we  find  p^0  in  mm.  = 
1489  and  pll9  in  mm.  =  1073.7.  Hence,  according  to  the  ex- 
planation given  before,  in  order  to  transform  these  pressures 
into  kg.  per  square  m.  we  have  to  divide  by  760  and  mul- 
tiply by  10,333.  We  obtain 


(J)    =  (1489  ~  yWH*  =  564.64. 

It  is  seen,  therefore,  that  the  two  laws  of  thermodynamics 
in  connection  with  the  experimental  data  on  saturated  water- 
vapor  enable  us  to  calculate  all  the  quantities  which  enter 
into  the  expression  for  the  external  work  done  by  the  adia- 
batic  expansion  of  such  a  vapor. 

It  should  be  noticed  that  this  expression  is  very  much 
different  from  the  corresponding  expression  obtained  for  the 
adiabatic  expansion  of  a  perfect  gas.  This  difference  would 
not  exist  if  saturated  vapors  obeyed  the  law  of  Mario  tte-Gay- 
Lussac. 

The  following  table  is  taken  from  Clausius*;  its  verifica- 
tion is  recommended  as  a  very  useful  exercise.  It  contains  the 
external  work  per  kg.  of  saturated  vapor  when  the  initial 
temperature  is  t  =  150°  C.  and  the  initial  quantity  of  the 
liquid  part  is  zero,  hence  M  =  ?HV 

*  Mech.  Warmetheorie,  vol.  i.  p.  167. 


REVERSIBLE  CYCLES  IN  OASES  AND   VAPORS.      93 


t 

150° 

125° 

100° 

75° 

50° 

25° 

F 

TTZi 

0 

11,300 

23,200 

35,900 

49,300 

63,700 

Thus  the  external  work  done  by  a  kg.  of  saturated  water- 
vapor  in  expanding  adiabatically  from  initial  temperature  of 
t  =  150°  C.  until  its  temperature  has  sunk  down  to  100°  C. 
is  23200  kg.  meters. 

CBS.  1.  In  evaporating  a  kg.  of  water  at  150°  to  a  kg.  of 
steam  at  150°  work  must  be  done  against  the  vapor  pressure 
corresponding  to  that  temperature.  This  work  can  be  easily 
found  to  be  18,700  kg.  meters. 

OBS.  2.  In  the  above  calculation  Clausius  assumed  J^ 
423.55. 

(y)  External  Work  done  during  Isothermal  Expansion. 

During  isothermal  expansion  pressure  remains  constant, 
since  temperature  remains  constant.  The  heat  supplied  is 
utilized  to  compensate  two  distinct  processes  which,  taken 
together,  constitute  the  operation  of  isothermal  expansion. 
The  two  processes  are:  separation  of  the  vapor  from  the 
liquid  mass,  and  overcoming  of  the  external  pressure.  Ac- 
cording to  our  definition  of  p,  that  is,  the  latent  heat  of 
evaporation,  the  total  heat  necessary  to  evaporate  m  kilo- 
grammes of  vapor  is  mp.  That  part  of  this  total  heat  which 
does  the  external  work  is  given  by 


W  =  fpdv. 


94  THERMODYNAMICS   OF 

But  v  =  mu  -j-  M<r,  and  in  isothermal  expansion 


dv  =  -—  dm  =  udm. 


/m 
pudm  =  pum, 
_ 


since  p  and  u  remain  constant  during  this  expansion 

We  are  ready  now  to  take  up  the  discussion  of  a  complete 
cycle  of  operations  upon  water-vapor. 


THE  INDICATOR  DIAGRAM  OF  A  SIMPLE  CYCLE. 

We  shall  arrange  the  cycle  in  such  a  way  that  it  shall  ap- 
proach as  nearly  as  possible  the  cycles  which  occur  in  actual 
steam-engines,  without  introducing  the  difficult  questions 
concerning  the  various  causes  which  produce  considerable 
differences  between  the  results  obtained  by  pure  theory  and 
the  results  obtained  from  the  actual  operation  of  the  various 
types  of  steam-engines.  The  existence  of  these  differences  is 
no  proof  of  the  weakness  of  the  fundamental  features  of  our 
theory.  On  the  contrary,  these  very  differences  prove  its 
strength,  for  they  point  out  the  important  fact  that  some  of 
the  changes  or  phenomena  which  take  place  during  the  cyclic 
operations  of  actual  steam-engines  have  not  been  taken  into 
account  in  our  theoretical  calculations.  Careful  experimental 
investigations  of  these  ignored  phenomena  and  the  study  of 
their  bearing  upon  the  performance  of  the  steam-engine,  not 
only  mean  the  gradual  extension  towards  a  complete  theory 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.       95 

of  the  steam-engine,  but  they  also,  by  widening  out  that 
theory,  bring  us  nearer  and  nearer  to  a  perfect  steam-engine. 
Fig  9  represents  symbolically  the  essential  parts  of  a 
modern  steam-engine.  A  is  the  boiler,  maintained  at  con- 
stant temperature  Tl  by  the  action  of  the  heat  supplied  by 
fuel.  C  is  the  condenser  maintained  at  constant  tempera- 


FIG.  9. 


ture  T0.  To  simplify  our  discussion,  we  shall  suppose  that 
the  condenser  is  maintained  at  constant  temperature  by 
taking  away  all  the  heat  that  may  be  supplied  to  it  during 
the  cycle  which  we  are  considering,  and  not  by  injection  of 
cold  water.  E  is  the  cylinder  in  which  a  piston  can  glide  up 
and  down  without  friction.  We  shall  suppose  that  the  cylin- 
der is  impermeable  to  heat.  The  action  of  the  pump  D  will 
be  explained  presently.  We  shall  now  suppose  that  by  special 
mechanical  contrivances  we  can  at  any  moment  connect  the 


96  THERMODYNAMICS  OF 

cylinder  to  or  disconnect  it  from  either  the  boiler  A  or  the 
condenser  C. 

Operation  1. 

At  the  beginning  of  the  cycle  the  piston  is  at  the  bottom  of 
the  cylinder.  Steam  is  now  admitted  from  the  boiler,  and 
when  its  quantity  is  ml  the  connection  with  the  boiler  is  cut 
off.  Suppose  that  a  quantity  of  the  liquid  equal  to  M  —  mt 
also  enters  with  the  steam.  The  piston  goes  up  until  the 
volume  v^  in  the  cylinder  is 


Since  the  pressure  during  this  operation  is  constant  and 
equal  to  pl9  (the  saturated-vapor  tension  at  the  initial  tem- 
perature TJ,  the  external  work  Wl  done  during  this  operation 
is  given  by 

Wl=pl(mlul+  M<r). 

The  quantity  of  heat  Q1  consumed  during  this  isothermal 
expansion  is  m^. 

In  order  to  represent  the  cycle  graphically,  refer  the  press- 
ures on  the  piston  and  the  volume  in  the  cylinder  to  a  set  of 
rectangular  axes  OA  and  OB,  taking  the  pressures  in  kg.  per 
sq.  m.  for  ordinates  and  the  corresponding  volumes  in  cub.  m. 
for  abscissae  (Fig.  10*). 

*This  diagram  was  taken  from  an  actual  steam-engine.  Boiler 
gauge,  165  ;  vacuum  gauge,  12.75  ;  rev.  per  minute,  76  ;  back  pressure, 
17  Ibs. 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.       97 

The  first  operation  will  be  represented  by  the  isothermal 
line  ab.     The  length  of  db  =  Ob'  represents  the  numerical 


0 


FIG.  10. 


value  of  vl  =  m1u1  -J-  Mcr.     The  area  aWO  is  numerically 
equal  to  Wr 

Operation  2. 

Disconnect  the  cylinder  from  the  boiler,  and  by  gradually 
diminishing  the  pressure  let  the  steam  expand  from  this  point 
on  adiabatically  until  its  temperature  has  sunk  to  T^  the 
temperature  of  the  condenser.  During  this  operation  the 
steam,  as  was  explained  before,  not  only  remains  saturated 
but  even  condenses  rapidly.  The  adiabatic  curve  will  be 
somewhat  like  the  curve  be.  Its  exact  form  can  be  traced  very 
accurately  by  theoretical  calculation,  but  since  this  is  a  some- 
what tedious  process  we  shall  omit  it  from  our  discussion. 
The  external  work  done  during  this  operation  is  represented 
by  the  area  Icc'b'.  Denoting  it  by  W\  we  shall  have,  as  has 
been  shown  before, 


-  mlulp1  - 


-  Mc(Tl  -  Ta).  j. 


98  THERMODYNAMICS  OF 


Operation  3. 

Connect  the   cylinder  with  the  condenser  and  compress. 
The  initial  volume  in  the  cylinder  is 


Let  the  connection  with  the  condenser  be  maintained  until 
the  piston  reaches  its  initial  position  again.  Since  the 
pressure  during  this  operation  remains  constant  and  equal 
to  j9a,  the  work  done  during  this  operation  will  be,  denoting  it 
by  W9, 


The  curve  of  compression  cd  is  an  isothermal  ;  it  is  evidently 
a  straight  line  parallel  to  OB.  The  area  CG'  Od  represents 
numerically  the  value  of  W3. 

This  operation  completes  the  cycle  as  far  as  the  process  in 
the  cylinder  is  concerned.  The  diagram  abed  is  called  the 
indicator  diagram  of  the  cycle.  Its  area  is  numerically  equal 
to  the  external  work  done  during  the  cyclic  process.  Let 
this  be  W,  then 

W=  W,+  W%+  W, 

KM,?,-  mlulpl  -  J{  (mjr%  -  m,r}) 


=  M<r(Pl  -  p,)  +  JMc(T,  -  Tt)  - 


REVERSIBLE  CYCLES  IN  OASES  AND    VAPORS.       99 

Corrections  for  the  Work  expended  upon  the  Pump. 

The  cycle  given  above  is  not  complete  because  the  state  of 
the  boiler  and  condenser  is  not  the  same  at  the  end  of  the 
cycle  as  in  the  beginning  of  it.  In  the  first  place,  the  quantity 
of  the  water  in  the  condenser  is  increased  and  that  in  the 
boiler  diminished  by  M  kilogrammes.  To  make  the  cycle 
complete  it  is  necessary  now  to  transfer  this  quantity  M  from 
the  condenser  to  the  boiler  by  the  action  of  the  pump.  The 
work  of  the  pump  subtracted  from  W  gives  us  then  the  avail- 
able external  work  of  the  complete  cycle. 

The  work  of  the  pump  is  easily  calculated  thus  : 
In  order  to  transfer  the  quantity  M  from  the  condenser  to 
the  pump  the  piston  of  the  pump  must  go  up  until  it  has  in- 
creased the  available  volume  in  the  cylinder  of  the  pump  by 
M(T.  During  the  upward  stroke  of  the  latter's  piston  a  valve 
connects  the  pump  to  the  condenser,  and  the  condenser  press- 
ure p^  forces  the  quantity  M  into  the  cylinder.  The  work  of 
the  pressure  p^  will  evidently  be 


The  valve  connecting  the  pump  to  the  condenser  is  then 
turned  off,  and  another  valve  is  turned  on  which  connects  the 
pump  to  the  boiler.  The  piston  of  the  pump  is  now  pressed 
down  and  the  quantity  M  of  the  fluid  is  forced  into  the  boiler 
against  the  pressure  pr  This  work  is  evidently 


W=  — 


100  THERMODYNAMICS  OF 

Adding  JF4  -f-  Wb  to  JFwill  give  us  W,  that  is,  the  available 
external  work  done  during  the  complete  cycle.     Hence 

W  =  JMc(T,  -  T,)  -  J(m,r9  -  m^). 

The  quantity  of  heat  expended  so  far  to  gain  this  amount  of 
external  work  is  m^,.  But  this  is  not  all.  For  in  order  to 
establish  the  same  state  of  the  boiler  and  condenser  as  at  the 
beginning  of  the  cycle  it  is  necessary  to  heat  the  quantity  M 
of  the  liquid,  after  it  enters  the  boiler  again,  from  the  tem- 
perature Tt  to  the  temperature  Tl9  for  which  an  additional 
quantity  of  heat,  MC(Tl  —  T9),  is  required.  We  have  there- 
fore on  the  whole  a  gain  of  W  units  of  available  external 
work  at  the  expense  of  (very  nearly)  J{m1r1  -j-  Mc(Tl  — 
units  of  heat. 


W 

— 7=rr~  —  efficiency  = 


or 


,  _ 


Calculation  of  the  Efficiency  in  Terms  of  the  Initial  and 
Final  Temperatures. 

In  order  to  obtain  a  more  suggestive  comparison  of  this 
result  to  the  result  which  we  obtained  in  the  case  of  a  simple 
reversible  cycle  with  a  perfect  gas,  it  is  desirable  to  express  E 
in  terms  of  initial  and  final  temperature.  The  second  opera- 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.     101 

tion   being   an   adiabatic   expansion,  we  have  the  following 
relation  : 


Substituting  this  value  of  wara  in  the  expression  for  E9  we 
obtain 


War,  -  T,  +  T,  log  J- 

_        \     __  •*•  1 


Mc(Tt  -  T,)  +  mfl 

To  simplify  this  equation  somewhat,  suppose  that  at  the  be- 
ginning of  the  cycle  no  part  of  the  mass  is  liquid.    Hence 

ml  =  M, 

and 


TlC(l  +  -n^-nr  log  w 

_    _  V  ^  1  ^2  ±\ 


T  —  T 

•*•!  *« 


/      ^  2"\ 

It  can  be  easily  shown  that  (       _x       log  ^j  is  always 


greater  than  unity  as  long  as  T9  <  T^.  Hence  the  efficiency 
of  our  cycle  is  smaller  than  that  of  a  simple  reversible  cycle 
performed  upon  a  perfect  gas.  Now  the  efficiency  of  a 


102  THERMODYNAMICS  OF 

simple  reversible  cycle  is  independent  of  the  substance 
operated  upon,  hence  we  are  forced  to  admit  that  the  cycle 
described  above  is  not  a  simple  reversible  cycle.  A  moment's 
consideration  will  convince  you  that  it  evidently  cannot  be 
considered  as  such.  For  in  such  a  cycle  heat  should  be 
transmitted  between  bodies  of  equal  or  very  nearly  equal 
temperatures;  in  other  words,  there  should  be  at  no  moment 
during  the  cycle  a  contact  between  bodies  of  widely  different 
temperatures.  In  our  cycle  this  condition  was  not  fulfilled, 
for  the  temperature  of  the  feed-water  forced  into  the  boiler  by 
the  pump  was  not  supposed  to  be  the  same  as  that  of  the 
water  in  the  boiler.  The  operation  of  the  steam-engine  which 
we  discussed  is  as  near  an  approach  to  perfect  reversibility  as 
the  action  of  any  steam-engine  in  operation  to-day.  And  yet 
its  efficiency  is  below  that  of  a  Carnot  reversible  engine.  We 
traced  the  cause  of  this  to  the  fact  that  the  machine  does  not 
operate  by  perfectly  reversible  cycles,  and  since  no  types  of 
steam-engines  which  now  prevail  operate  by  perfectly  reversi- 
ble cycles,  it  follows  that  the  efficiency  of  the  now  prevailing 
types  of  steam-engines  would  even  under  ideal  conditions  be 
lower  than  that  of  Carnot' s  reversible  engine. 

This  remarkable  fact  is  but  a  single  illustration  of  a  general 
principle  first  announced  by  a  young  French  engineer,  Nicolas 
Leonhard  Sudi  Carnot,  in  his  immortal  essay,  "Reflexions 
sur  la,  puissance  mot  rice  du  feu  et  sur  les  machines  propres 
a  la  developper"  (Paris,  1824).  This  principle  may  be  stated 
as  follows: 


REVERSIBLE  CYCLES  IN  OASES  AND    VAPORS.    103 

Carnot's  Principle. 

Of  ah  engines  working  between  the  same  source  of  heat  and 
refrigeration  a  reversible  engine  gives  the  maximum  efficiency. 

This  principle  contains  Carnot's  solution  of  the  problem: 
"  What  are  the  conditions  under  which  the  maximum  work 
can  be  obtained  from  a  given  quantity  of  heat?" 

Add  to  this  principle  the  hypothesis  that  the  efficiency  of 
a  reversible  engine  is  independent  of  the  material  operated 
upon,  or,  what  amounts  to  the  same  thing,  that  the  efficiency 
of  all  reversible  engines  is  the  same,  and  the  second  law  of 
thermodynamics  in  all  its  generality  follows  at  once.  But 
this  very  hypothesis  may  be  said  to  be  the  fundamental 
hypothesis  of  Carnot.  There  is  no  doubt,  therefore,  that  to 
this  great  Frenchman  belongs  the  glory  of  having  discovered 
one  of  the  most  important  laws  in  the  whole  range  of  physi- 
cal sciences. 

If  the  first  law  of  thermodynamics  be  defined  as  the  prin- 
ciple of  equivalence  between  heat  and  mechanical  work  and 
not  as  a  particular  case  of  the  Principle  of  Conservation  of 
Energy,  then,  according  to  historic  evidences  which  we 
possess  to-day,  Carnot  is  certainly  the  discoverer  of  the  first 
law  as  well.  Observe  here  that  from  this  principle  of 
equivalence  to  the  general  principle  of  conservation  of 
energy  there  are  but  few  and  easy  steps.* 

It  is  an  established  historical  fact  that  Carnot  had  made 
the  necessary  preparations  to  verify  experimentally  his  ideas 
on  the  two  fundamental  laws  of  the  theory  of  heat  when  a 

*  See  notes  following  the  edition  of  1878  of  his  great  work,  Re- 
flexions, etc.  (Gauthier-Villars,  Paris). 


104  THERMODYNAMICS  OF 

premature  death  deprived  the  world  of  one  of  its  most  brill- 
iant intellects.  Carnot  died  in  1832,  when  only  36  years  of 
age. 

Reversible  Steam-engine. 

We  shall  now  close  onr  discussion  of  thermodynamics  of 
saturated  vapors  by  describing  a  simple  reversible  process 
performed  with  saturated  water-vapor.  This  will  give  us  a 
pretty  faithful  picture  of  the  possible  type  of  a  perfectly 
reversible  steam-engine;  it  will  also  afford  us  an  oppor- 
tunity of  showing  by  actual  calculation  that  a  Carnot  engine 
when  operating  upon  a  body  which  is  not  a  perfect  gas  has 
the  same  efficiency  as  when  it  operates  upon  such  a  gas. 

Place  a  quantity  of  ml  kilogrammes  of  water  of  temper- 
ature Tl  into  the  cylinder  of  a  Carnot  engine.  Let  there  be 
two  large  reservoirs  A  and  B  of  temperatures  7\  and  7'2  re- 
spectively. Let  the  piston  be  in  contact  with  the  liquid  at 
the  beginning  of  the  operation. 

Connect  the  cylinder  to  A  and  allow  the  piston  to  go  up 
slowly.  There  will  be  evaporation,  but  the  temperature  and 
consequently  the  pressure  also  will  remain  constant,  because 
A  supplies  the  heat  of  evaporation.  Let  the  evaporation 
continue  until  all  the  water  is  evaporated.  Denote  the  heat 
supplied  by  A  during  this  operation  by  Ql  and  the  external 
work  done  by  TT1?  then 

Qi  =  mlpl     and     Wl  =  p^m^ 

Disconnect  the  reservoir  and  let  the  steam  expand  adiabat- 
ically  until  the  temperature  7\  of  the  reservoir  B  is  reached 


REVERSIBLE  CYCLES  IN  OASES  AND    VAPORS.    105 

The  steam  will  remain  saturated  on  account  of  condensation. 
The  external  work  JF2  done  daring  this  operation  will  be 

W,  =  /W»,  —  /W»i  +  m*Pi  -  ™*P*  +  7 

Connect  now  to  reservoir  B  and  compress  until  the  weight 
of  steam  is  reduced  to  m9.  Temperature  and  therefore  pres- 
sure remain  constant  during  this  isothermal  operation.  The 
external  work  W3  done  during  this  operation  will  be 

W9  =    ipdv. 

Buu  since  v  =  mu  -±-m.cr    and    dv  =  —  -  dm  =  udm, 

3m 

/ma 
pudm  =  p^u^(mz  —  m2). 


The  quantity  ma  must  be  such  that  the  next  adiabatic  opera- 
tion shall  be  capable  of  reducing  the  whole  quantity  to  liquid 
at  T1  degrees. 

By  disconnecting  the  cylinder  from  the  reservoir  B  we 
compress  adiabatically.  The  steam  will  condense  and  the 
temperature  will  rise.  But  since  the  steam  is  continually  ia 
contact  with  its  own  liquid  it  will  remain  saturated.  At  the 
end  of  this  operation  we  shall  have  again  ml  kg.  of  water  at 
T1  degrees.  The  work  done  during  this  operation,  denoting 
it  by  TF4  ,  can  be  calculated  from  the  relation  which  we  de- 
duced before,  namely, 

W  =  pum  —  p&jn^  +  w3p2  —  mp  + 


106  THERMODYNAMICS  OF 

but  we  must  remember  that  the  weight  m  of  steam  at  the  end 
of  our  operation  is  zero;  hence 


Denote  the  total  external  work  during  this  cycle  by  W', 
then 


^  T,)] 


The  expressions  in  the  brackets  stand  for   the  various  W's. 
This  expression  reduced  gives 

W  =  (m,  -  m,)p,  +  mfa 

We  can  express  W  in  terms  of  the  initial  and  final  tem- 
perature by  the  relation  deduced  in  our  discussion  of  the 
condensation  during  adiabatic  expansion,  viz., 


T 

,  =  -  ?X  log     2; 


This  substituted  in  the  expression  for  W  gives 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.     107 
The  efficiency  of  the  cycle  is,  according  to  definition, 

fjj  rri 

i  -^  2 

*£  i  i/i  l 

The  efficiency,  therefore,  is  just  the  same  as  that  obtained 
by  operating  in  a  reversible  manner  upon  a  perfect  gas.  It 
is,  therefore,  higher  than  the  efficiency  of  the  cycles  of  any 
other  type  of  steam-engine. 

But  why,  then,  should  the  now  prevailing  types  of  steam- 
engines  be  so  widely  different  from  Carnot's  type  if  a  devia- 
tion from  this  type  brings  with  it  a  loss  of  efficiency  ?  The 
complete  answer  to  this  question  will  be  given  in  the  course 
on  mechanical  engineering.  Let  two  suggestions  suffice  here: 
first,  efficiency  is  not  the  only  guiding  consideration  in  the 
construction  of  steam-engines,  or  for  that  matter  in  the  con- 
struction of  any  other  kind  of  engines.  The  question  of  out- 
put is  a  very  serious  matter  too,  and  in  order  to  increase  the 
output  we  must  very  often  make  some  sacrifice  in  the 
efficiency.  Secondly,  the  difference  between  the  efficiency 
of  the  now  preva.ling  types  of  steam-engines  and  that  of  the 
ideal  type  just  described  is  due  far  more  to  other  things  than 
to  the  fact  that  the  operations  which  we  discussed  above 
are  not  perfectly  reversible.  Consider  the  losses  due  to 
radiation,  the  action  of  the  conductingc  ylinder-walls  on 
the  expanding  steam,  etc.,  etc.  So  that  the  completeness  of 
tho  reversibility  of  the  operations  through  which  the  steam 
passes  in  these  engines  is  not  quite  as  important  a  matter  as 
it  might  appear  at  first  sight, 


108  THERMODYNAMICS  OF 

SUMMARY. 

1.  Method  of  Reasoning  which  led  to  the  General  Form  of  the 
Second  Law  of  Thermodynamics. 

After  discussing  the  application  of  the  first  law  of  thermo- 
dynamics to  the  study  of  reversible  operations  performed 
upon  perfect  gases  we  passed  on  to  a  similar  discussion  for 
the  next  simplest  class  of  bodies,  that  is,  saturated  vapors. 
We  suggested,  however,  that  it  would  be  advisable  to  deduce 
first  another  general  law  which,  together  with  the  first  law, 
forms  the  foundation  of  the  science  of  thermodynamics:  we 
mean  the  second  law.  In  doing  that  we  followed  practically 
in  the  steps  of  the  great  Carnot.  For  he  had  to  discover 
the  second  law  before  he  was  able  to  pass  from  the  study  of 
the  reversible  gas-engine  to  that  of  the  steam-engine. 

The  first  and  simplest  form  of  the  second  law  was  obtained 
by  considering  the  efficiency  of  a  Carnot  engine  operating 
reversibly  upon  a  perfect  gas  by  simple  cycles.  This  form 
consisted  in  a  mathematical  statement  which  expressed  that 
the  efficiency  of  such  cycles  depends  on  the  extreme  tempera- 
tures of  the  cycles,  thus: 

Q  _Tt-T, 
Q,~        Tt 

We  then  introduced  the  axiom  of  Clausius  and  by  means  of 
this  axiom  we  showed  that  the  efficiency  of  a  simple  reversible 
cycle  is  independent  of  the  substance  operated  upon.  Hence 
it  is  always 

Q  _  T,  -  ra 


REVERSIBLE  CYCLES  IN  GASES  AND    VAPORS.     109 
We  then  transformed  this  relation  into  the  following  : 


It  was  stated  that  this  transformed  expression  is  preferable 
because  it  lends  itself  more  readily  to  generalization.  In 
order  to  effect  this  generalization  we  passed  on  to  the  con- 
sideration of  a  complex  reversible  cycle  and  obtained  the  rela- 
tion 

~m      \     ~7rT   T   flS      T    •    •    •    ~T  ~m~~  ~=  ®9 


or 


'T: 

We  finally  considered  a  reversible  cycle  of  infinite  com- 
plexity and  obtained 

C*Q_n 

J  T 

where  the  integral  is  to  be  extended  all  around  the  closed 
curve  which  represents  graphically  the  reversible  cycle  of 
infinite  complexity,  that  is  to  say,  all  around  the  indicator 
diagram. 

This  is  the  integral  form  of  the  second  law  for  reversible 
cycles.  But  inasmuch  as  the  integral  laws  in  general  describe 
resultant  effects  only  of  a  series  of  physical  processes  and 
throw  but  very  little  light  upon  each  process  of  that  series,  \vo 

^ 


110  THERMODYNAMICS  OF 


proceeded  to  deduce  from  the  integral  law  /  -~  =  0  a  dif- 
ferential law,  that  is  to  say  a  relation  which  will  hold  true 
during  every  interval,  no  matter  how  small,  of  the  entire  time 
during  which  the  cyclic  process  is  completed.  We  obtained 


That  is  to  say,  the  heat  absorbed  by  a  body  during  any  one 
of  the  infinite  number  of  infinitely  small  operations  which 
taken  together  form  the  complete  reversible  cycle  is  equal  to 
the  absolute  temperature  of  the  body  multiplied  by  the  incre- 
ment of  the  entropy  of  the  body.  The  definition  of  the 
entropy  being :  1st.  Mathematical  and  complete.  It  is  a  finite, 
continuous,  and  singly-valued  function  of  the  co-ordinates 
which  define  the  state  of  the  body,  that  is,  of  pressure,  volume, 
and  temperature,  which  satisfies  the  above  differential  equa- 
tion for  any  reversible  operation.  2d.  Physical  meaning.  Its 
variation  during  an  isothermal  operation  is  equal  to  the  heat 
absorbed  or  given  off,  divided  by  the  absolute  temperature  at 
which  the  operation  takes  place.  Its  variation  is  zero  during 
an  adiabatic  operation. 

2.  Forms  of  the  Two  Laws  in  the  case  of  Reversible  Opera- 
tions on  Saturated  Vapors. 

Starting,  then,  with  our  two  laws  of  thermodynamics, 
dQ  =  dU  +  pdv, 
dQ  =  TdS, 


EEVERSIBLE  CYCLES  IN  GASES  AND   VAPORS.    Ill 

we  took  up  the  discussion  of  saturated  vapors,  and  in  this 
discussion  we  followed  up  as  much  as  possible  the  line  of 
reasoning  which  we  employed  in  the  discussion  of  reversible 
processes  in  perfect  gases. 

1st.  We  discussed  the  most  essential  physical  properties  of 
saturated  vapors  by  consulting  carefully  the  records  of  physi- 
cal research. 

2d.  We  then  performed  two  simple  processes  for  the  purpose 
of  defining  the  physical  constants  of  a  saturated  vapor.  These 
two  processes  we  called  the  definitional  processes  (A)  and  (B). 
They  gave  us  the  following  two  definitional  relations: 


. 

9m 


These  two  definitional  relations  enabled  us  then  to  express  the 
two  laws  of  thermodynamics  in  the  case  of  saturated  vapors 
in  a  very  convenient  form,  viz., 


and  by  combining  these  two  we  obtained  a  third  very  con- 
venient relation; 


112  THERMODYNAMICS  OF 

By  introducing  the  kg.  calorie  as  the  unit  of  heat  (as  is  gener- 
ally done  in  experimental  researches)  we  obtained 


(II-) 


3.  Application  of  the  Two  Laws  to  the  Study  of  the  Physical 

Constants  of  Saturated  Vapor. 

We  then  proceeded  to  apply  these  fundamental  thermo- 
dynamical  relations  to  the  study  of  saturated  vapors,  and  in 
particular  to  the  study  of  saturated  water-vapor.  We  divided 
this  discussion  into  three  parts.  One  part  related  to  the  dis- 
cussion of  the  constants  c,  h,  r,  u,  and  p,  especially  to  the 
methods  of  calculating  two  of  them,  h  and  u,  when  the  other 
three  are  given  by  experimental  data.  The  negative  value  of 
the  specific  heat,  the  increase  of  density  with  the  temperature, 
and  the  deviation  of  the  behavior  of  saturated  water- vapor 
from  the  law  of  Mariotte-Gay-Lussac  were  discussed  with 
particular  emphasis. 

4.  Application  of  the  Two  Laws  to  the  Study  of  Isothermal 
and  Adiabatic  Expansion  of  Saturated  Water-vapor. 

Following  the  method  of  discussion  which  we  employed  in 
the  case  of  perfect  gases  we  then  took  up  the  very  important 


REVERSIBLE  CYCLES  IN  GASES  AND   VAPORS.    113 

part  of  our  work  in  thermodynamics,  that  is,  adiabatic  and 
isothermal  expansion  of  saturated  water-vapor,  the  condensa- 
tion which  takes  place  during  adiabatic  expansion  and  the 
superheating  wliicli  takes  place  during  adiabatic  compression 
were  particularly  dwelt  upon.  The  external  work  done  dur- 
ing adiabatic  and  isothermal  operations  received  our  most 
careful  attention. 


5.  Non-reversible  and  Reversible  Cycles  of  Operation  upon 
Saturated  Water-vapor. 

"We  were  ready  then  to  take  up  the  question  of  the  theo- 
retical efficiency  of  the  non-reversible  and  the  reversible  types 
of  steam-engines.  This  question  is,  from  an  engineering 
standpoint,  the  most  important  of  all  questions  in  thermo- 
dynamics. It  is  also  exceedingly  important  from  a  historical 
point  of  view,  for  the  study  of  this  question  led  Carnot  to  the 
discovery  of  the  two  laws  of  thermodynamics. 

The  external  work  done  during  each  component  operation 
of  these  cycles  was  calculated  and  expressed  by  formulae  which 
admit  of  being  reduced  to  actual  figures,  so  that  the  action  of 
each  component  operation  could  be  studied  by  finding  the 
numerical  value  of  this  action  in  terms  of  kg.  meters.  The 
theoretical  efficiency  of  non-reversible  machines  was  then 
shown  to  be  smaller  than  that  of  the  reversible  type,  and  the 
efficiency  of  this  was  then  demonstrated  by  actual  calculation 
to  be  the  same  as  that  of  a  Carnot  reversible  engine  operating 
by  simple  reversible  cycles  upon  a  perfect  gas. 

There  remains  still  another  important  application  of  the 
two  laws  of  thermodynamics  which  is  within  the  limits  of 


114  THEUMOD  TNAMICS. 

this  purely  theoretical  course.  It  is  the  application  to  the 
study  of  the  flow  of  gases  and  vapors.  In  this  part  of  our 
discussion  we  propose  to  follow  closely  the  beautifully  worked 
out  Chapter  IX  in  Peabody's  "  Thermodynamics  of  the  Steam- 
engine." 

These  are  the  most  essential  elements  of  theoretical  thermo- 
dynamics, upon  which  as  a  strong  foundation  the  practical 
engineer  must  raise  the  vast  structure  of  the  beautiful  science 
of  Caloric  Engineering. 


. 


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